Efoevi Angelo Koudou

Jacek Wesolowski

Jacek Wesolowski Independence

Angelo Efo

Évi Koudou

Jacek Weso Lowski

INDEPENDENCE PRESERVING PROPERTY OF KUMMER LAWS

We prove that if X, Y are positive, independent, non-Dirac random variables and if for α

 is necessarily a product measure with Kummer marginals. The result extends earlier characterizations of Kummer and gamma laws by independence of

1+X , which corresponds to the case of ψ 1,0 .

We also show that this independence property of Kummer laws covers, as limiting cases, several independence models known in the literature: the Lukacs, the Kummer-Gamma, the Matsumoto-Yor and the discrete Korteweg de Vries models.

Introduction

Consider, for b, c > 0, the gamma distribution Gamma(b, c) with density proportional to y b-1 e -cy I (0,∞) (y) and for p ∈ R, a > 0, b > 0, the generalized inverse Gaussian (GIG) distribution GIG(p, a, b) with density proportional to

x p-1 e -ax-b/x I (0,∞) (x).

Following [START_REF] Croydon | Detailed balance and invariant mesures for discrete KdV-and Toda-type systems[END_REF], we say that a quadruplet of probability measures (µ, ν, μ, ν) on U, V, Ũ, Ṽ, respectively, satisfy the detailed balance equation for a map F :

U × V → Ũ × Ṽ if F (µ ⊗ ν) = μ ⊗ ν,
where

F (µ ⊗ ν) means (µ ⊗ ν) • F -1 .
The Matsumoto-Yor property is the following: for p, a, b > 0, given two independent, positive random variables X and Y such that X ∼ GIG(-p, a, b) and Y ∼ Gamma(p, a), the random variables 1 X+Y and 1

X -1 X+Y are independent (and follow the GIG(-p, b, a) and Gamma(p, b), respectively). Using the terminology of [START_REF] Croydon | Detailed balance and invariant mesures for discrete KdV-and Toda-type systems[END_REF], the Matsumoto-Yor property says that the quadruplet of probability measures µ = GIG(-p, a, b), ν = Gamma(p, a), μ = GIG(-p, b, a), ν = Gamma(p, b) satisfy the detailed balance equation for the map F : (0, ∞) 2 → (0, ∞) 2 (x, y)

F → 1 x + y , 1 x - 1 x + y .
This property was discovered in [START_REF] Matsumoto | An analogue of Pitman's 2M -X theorem for exponential Wiener functionals. Part II: the role of the generalized inverse Gaussian laws[END_REF] in the case a = b, while studying some functionals of exponential Brownian motion. In [START_REF] Letac | An independence property for the GIG and gamma laws[END_REF] it was noticed that it holds also when a ̸ = b and it was proved that this independence property is in fact a characterization: for two non-Dirac, positive and independent random variables X and Y , the random variables 1 X+Y and 1 X -1 X+Y are independent if and only if X ∼ GIG(-p, a, b) while Y ∼ Gamma(p, a) for some p, a, b > 0.

In [START_REF] Koudou | Independence properties of the Matsumoto-Yor type[END_REF] the authors studied the question of finding decreasing and bijective functions f : (0, ∞) → (0, ∞) such that there exists a quadruplet of probability measures (µ, ν, μ, ν) on (0, ∞) satisfying the detailed balance equation for the map T f : (0, ∞) 2 → (0, ∞) 2 (x, y) → (f (x + y), f (x) -f (x + y)).

This led, at the cost of some regularity assumptions, to other independence properties of Matsumoto-Yor type (of course, one retrieves the original Matsumoto-Yor case for f (x) = 1/x), amongst which was a property involving the Kummer distribution: for a, c > 0 and b ∈ R, the Kummer distribution K(a, b, c) is defined through the density proportional to

x a-1 e -cx (1+x) b I (0,∞) (x). More precisely, it was proved in [START_REF] Koudou | Independence properties of the Matsumoto-Yor type[END_REF] that if X and Y are independent Kummer and gamma random variables with suitably related parameters then

U = X + Y and V = 1+(X+Y ) -1 1+X -1
are independent Kummer and beta random variables. This was the starting point of a number of works on Matsumoto-Yor type characterizations of the Kummer distribution. Firstly, starting from the latter property and looking for an involutive version of it, i.e. trying to find an involutive map F : (X, Y ) → (U, V ) for which the Kummer distribution is involved in a detailed balance equation, the following interesting property was discovered in [START_REF] Hamza | On Kummer's distribution of type two and a generalized beta distribution[END_REF]: let X and Y be independent, X have the distribution K(a, b, c) and Y have the gamma distribution Gamma(b, c), then [START_REF] Abramowitz | Pocketbook of Mathematical functions[END_REF] U = Y 1+X and V = X 1+X+Y 1+X are independent, U ∼ K(b, a, c) and V ∼ Gamma(a, c).

In [START_REF] Piliszek | Change of measure technique in characterizations of the Kummer and gamma laws[END_REF] this independence property was proved to give a characterization result with no assumption of existence of densities. Related characterizations were considered in [START_REF] Weso Lowski | On the Matsumoto-Yor type regression characterization of the gamma and Kummer distributions[END_REF] and [START_REF] Piliszek | Kummer and gamma laws through independencies on trees -another parallel with the Matsumoto-Yor property[END_REF]. In [START_REF] Ko Lodziejek | Independence characterization for Wishart and Kummer random matrices[END_REF] an extension to the matrix-variate case was established, while in [START_REF] Piliszek | Regression conditions that characterize free-Poisson and free-Kummer distributions[END_REF] a free probability version of the property and characterization was given. The latter needed a definition of a new distribution, a free analogue of the Kummer distribution.

In the past ten years there has been a revival of interest in transformations preserving independence properties, triggered mostly by invariance properties of random iterations schemes or random walks in random environment. In particular, the log-gamma random polymer introduced in [START_REF] Seppäläinen | Scaling for a one-dimensional directed polymer with boundary conditions[END_REF] relies on the Lukacs independence property for the Gamma distributions (see [START_REF] Lukacs | A characterization of the gamma distribution[END_REF]), while the beta polymer introduced in [START_REF] Barraquand | Random walk in beta-distributed random environment[END_REF] refers to an independence property for beta distributions, called neutrality. This property can be traced back to the characterization of the bivariate Dirichlet distribution of [START_REF] Darroch | A characterization of the Dirichlet distribution[END_REF] (see also [START_REF] Seshadri | Constancy of regressions for beta distributions[END_REF]). For a complete description of all stationary polymers on the lattice Z 2 + see [START_REF] Chaumont | Characterizing stationary 1 + 1 dimensional lattice polymer model[END_REF], where also many further references can be found.

Let us come back to the definition of the detailed balance equation and recall its context as described in [START_REF] Croydon | Detailed balance and invariant mesures for discrete KdV-and Toda-type systems[END_REF], which considers models assuming the following dynamics: for (n, t) in Z 2 , n is the spatial coordinate and t the temporal one. For fixed t ∈ Z, (x t n ) n∈Z ∈ (0, ∞) Z is the configuration of the system at time t, and (y t n ) n∈Z ∈ (0, ∞) Z a collection of auxiliary variables through which the dynamics from t to t + 1 are defined. Namely, (x t n , y t n ) depends on (x t-1 n , y t n-1 ) only,

(2)

(x t n , y t n ) = G(x t-1 n , y t n-1 ),
where for a bijection F : X × Y → X × Ỹ either G = F , when n + t is even or G = F -1 when n + t is odd. The case when F is involutive is referred to as type I model, while the general case is referred to as type II model. Let x = (x n ) n∈Z be such that the above recursion with the initial condition x 0 n = x n , n ∈ Z, has a unique solution (x t n (x), y t n (x)) n,t∈Z . Let X * denote the set of all such x's. According to Theorem 1.1 in [START_REF] Croydon | Detailed balance and invariant mesures for discrete KdV-and Toda-type systems[END_REF], for a type I model, a sequence of iid random variables

X = (X n ) n∈Z with X 1 ∼ µ satisfies X d = x 1
n (X) n∈Z iff there exists a probability measure ν such that the pair (µ, ν) satisfies the detailed balance condition with respect to F . That is, µ ⊗ ν is the invariant measure for the recursion [START_REF] Bao | Characterizations of the generalized inverse Gaussian, asymmetric Laplace, and shifted (truncated) exponential laws via independence properties[END_REF]. In case of the type II model similar alternating invariance holds for pairs µ ⊗ ν and μ ⊗ ν depending on parity of n + t in [START_REF] Bao | Characterizations of the generalized inverse Gaussian, asymmetric Laplace, and shifted (truncated) exponential laws via independence properties[END_REF]. In [START_REF] Croydon | Detailed balance and invariant mesures for discrete KdV-and Toda-type systems[END_REF] the authors identified four such type I and/or type II models.

(1) Ultra-discrete KdV (Korteweg-de Vries): type I model with

F (x, y) := F (J,K) udK = (y -(x + y -J) + + (x + y -K) + , x -(x + y -K) + + (x + y -J) + )
with µ and ν the shifted truncated exponential or shifted scaled truncated geometric laws.

(2) Discrete KdV: type I model with

F (x, y) := F (α,β) dK (x, y) = y(1+βxy)
1+αxy , x(1+αxy) 1+βxy with µ the GIG law and ν the GIG (gamma) law which, when αβ = 0, has a direct connection with the Matsumoto-Yor property and related characterization of the GIG and gamma laws. (3) Ultra-discrete Toda: type II model with

F (x, y) := F udT * (x ∧ y, x -y)
with µ, ν, μ the shifted exponential, ν asymmetric Laplace or µ, ν, μ shifted scaled geometric, ν scaled discrete Laplace laws; this one is related to classical characterizations of the exponential and geometric distributions from [START_REF] Ferguson | A characterization of the exponential distribution[END_REF] and [START_REF] Crawford | Characterization of geometric and exponential distroibutions[END_REF]. (4) Discrete Toda: type II model with F (x, y) := F dT * (x, y) = x + y, x x+y with µ, ν, μ the gamma, ν the beta laws having a direct connection with the characterization of the gamma distribution given in [START_REF] Lukacs | A characterization of the gamma distribution[END_REF].

For relations to box-ball systems and Pitman's transform one can consult [START_REF] Croydon | Bi-infinite solutions for KdV-and Toda-type discrete integrable systems based on path encodings[END_REF] and [START_REF] Croydon | Dynamics of the box-ball system with random initial conditions via Pitman's tranform[END_REF]. Also recently a mysterious connection between independence properties and Yang-Baxter equations holding for related transformations was discovered in [START_REF] Sasada | Yang-Baxter maps and independence preserving property[END_REF].

In the context of the discrete KdV model in [START_REF] Croydon | Detailed balance and invariant mesures for discrete KdV-and Toda-type systems[END_REF] the authors observed that if X and Y are independent, U and V are independent and all four have GIG distributions with suitable parameters, then (X, Y ) and (U, V ) satisfy the detailed balance equation for the map F (α,β) dK . They also conjectured that the GIG distributions are the only possible ones that satisfy the F (α,β) dK -detailed balance equation. Recently, in [START_REF] Letac | About an extension of the Matsumoto-Yor property[END_REF] this conjecture was proved without the assumptions of existence and regularity of densities made in [START_REF] Bao | Characterizations of the generalized inverse Gaussian, asymmetric Laplace, and shifted (truncated) exponential laws via independence properties[END_REF] in proving a version of the same conjecture. More precisely, [START_REF] Letac | About an extension of the Matsumoto-Yor property[END_REF] established the following extension of the Matsumoto-Yor property: if A and B are non-degenerate, positive and independent random variables, and if α and β are two positive and distinct numbers, then the random variables

S = 1 B βA+B αA+B , T = 1 A βA+B αA+B
are independent if and only if A and B have GIG distributions with suitable parameters.

In this paper we reveal one more candidate for an invariant measure for a lattice recursion model. We derive the detailed balance equation for the Kummer distributions. Specifically, our main result (Theorem 4.3) gives a characterization of the Kummer laws, which is a result of a similar nature as the one in [START_REF] Letac | About an extension of the Matsumoto-Yor property[END_REF] for the GIG laws, i.e. it says that the Kummer distributions are the only possible ones which let the detailed balance equation be satisfied for the map (3) F (x, y) = y 1+β(x+y) 1+αx+βy , x 1+α(x+y) 1+αx+βy . The proof uses a suitably designed "Laplace-type" transform and leads to a special second order linear differential equation for an unknown function of such form. In this sense the general methodology (a Laplace type transform and a second order ordinary linear differential equation) resembles that of the proof from [START_REF] Letac | About an extension of the Matsumoto-Yor property[END_REF]. However, at the technical level, the challenges to overcome were of quite a different nature. Interpreting this result in the context of the lattice system of recursions described above, it says that the Kummer distributions are the only relevant invariant measures for the type I model governed by the F defined in [START_REF] Barraquand | Random walk in beta-distributed random environment[END_REF].

The paper is organized as follows: in Section 2 we introduce a scaled version of the Kummer distribution, we express and prove the considered independence property in terms of that scaled Kummer distribution (Theorem 2.1). We show in Section 3 that some limit versions of Theorem 2.1 cover several well-known independence preserving transformations. Indeed, relying on a version of Theorem 5.5 from [START_REF] Billingsley | Convergence of probability measures[END_REF], we prove that Theorem 2.1 yields, as limiting cases, the Lukacs property, [START_REF] Lukacs | A characterization of the gamma distribution[END_REF], the Kummer-Gamma property, [START_REF] Koudou | Independence properties of the Matsumoto-Yor type[END_REF], the Matsumoto-Yor property, [START_REF] Matsumoto | An analogue of Pitman's 2M -X theorem for exponential Wiener functionals. Part II: the role of the generalized inverse Gaussian laws[END_REF][START_REF] Letac | An independence property for the GIG and gamma laws[END_REF] and the KdV property, [START_REF] Croydon | Detailed balance and invariant mesures for discrete KdV-and Toda-type systems[END_REF]. In Section 4 we define and analyze the Kummer transform, an extended Laplace transform that will help us to prove the chacterization theorem formulated at the end of this section. Section 5 contains the proof of the characterization split in several steps (subsections), amongst which is the crucial observation that the unknown Kummer transform satisfies the Kummer differential equation (see [START_REF] Abramowitz | Pocketbook of Mathematical functions[END_REF]).

The independence property

For the purpose of this paper it will be convenient to introduce a scaled version of the Kummer distribution.

Definition 2.1. Let K α (a, b, c) for α ≥ 0, a, c > 0 and b ∈ R be the probability distribution defined by the density

f (x) ∝ x a-1 e -cx (1+αx) b I (0,∞) (x) Theorem 2.1. Assume that (X, Y ) ∼ K α (a, b, c) ⊗ K β (b, a, c) for a, b, c > 0 and α, β ≥ 0, α ̸ = β. Let (4) U = Y 1+β(X+Y ) 1+αX+βY and V = X 1+α(X+Y ) 1+αX+βY . Then (U, V ) ∼ K α (b, a, c) ⊗ K β (a, b, c).
Remark 2.1. Note that K 0 (a, b, c) = Gamma(a, c) and for α > 0 and X ∼ K α (a, b, c) we have αX ∼ K(a, b, c/α). Consequently, by taking (α, β) = (1, 0), we see that Theorem 2.1 is a straightforward extension of the independence property observed in [START_REF] Hamza | On Kummer's distribution of type two and a generalized beta distribution[END_REF] and recalled in the introduction, see [START_REF] Abramowitz | Pocketbook of Mathematical functions[END_REF].

Proof of Theorem 2.1. Denote ψ(x, y) = y 1+β(x+y) 1+αx+βy , x 1+α(x+y) 1+αx+βy =: (u, v), x, y > 0.
Note that ψ : (0, ∞) 2 → (0, ∞) 2 is an involution. Moreover, the following identities hold true:

x + y = u + v, (5)
x 1+βy = v 1+αu , (6)
y 1+αx = u 1+βv . (7)
Using ( 5), ( 6) and ( 7) we compute the Jacobian J

ψ -1 (u, v) of ψ -1 = ψ as follows (8) J ψ -1 (u, v) = ∂(x,y) ∂(u,v) = ∂x ∂u ∂x ∂v 1 -∂x ∂u 1 -∂x ∂v = ∂x ∂u -∂x ∂v = 1+αx+βy 1+αu+βv = xy uv .
Now we are ready to find the joint density of (U, V ). We have

f (U,V ) (u, v) = J ψ -1 (u, v) f X (x(u, v)) f Y (y(u, v)) ∝ xy uv x a-1 (1+αx) b e -cx y b-1 (1+βy) a e -cy I (0,∞) 2 (u, v) = 1 uv x 1+βy a y 1+αx b e -c(x+y) I (0,∞) 2 (u, v).
Again referring to ( 5), ( 6) and ( 7) we get

f (U,V ) (u, v) ∝ 1 uv v 1+αu a u 1+βv b e -c(u+v) I (0,∞) 2 (u, v)
and the result follows. □ Theorem 2.1 yields the following independence property for pure (i.e. α = β = 1) Kummer variables. In a very interesting recent paper, [START_REF] Sasada | Yang-Baxter maps and independence preserving property[END_REF], the authors announced (independently) a result (see their Theorem 1.1 (ii)), which is equivalent to the independence property observed in Theorem 2.1.

Corollary 2.2. Let ( X, Ỹ ) ∼ K(a, b, c) ⊗ K(b, a, γc) for 1 ̸ = γ > 0. Then ( Ũ , Ṽ ) := Ỹ X+γ(1+ Ỹ ) 1+ X+ Ỹ , X Ỹ +γ -1 (1+ X) 1+ X+ Ỹ ∼ K(b, a, c) ⊗ K(a, b, γc). Proof. For α, β ̸ = 0 denote X = αX, Ỹ = βY , Ũ = αU , Ṽ = βV . Let γ = α/β. Then α ̸ = β

Limiting cases of the Kummer independence property

We will show that Theorem 2.1 yields, as limiting cases, several well-known independence properties: the Lukacs property, [START_REF] Lukacs | A characterization of the gamma distribution[END_REF], the Kummer-Gamma property, [START_REF] Koudou | Independence properties of the Matsumoto-Yor type[END_REF], the Matsumoto-Yor property, [START_REF] Matsumoto | An analogue of Pitman's 2M -X theorem for exponential Wiener functionals. Part II: the role of the generalized inverse Gaussian laws[END_REF][START_REF] Letac | An independence property for the GIG and gamma laws[END_REF] and the KdV property, [START_REF] Croydon | Detailed balance and invariant mesures for discrete KdV-and Toda-type systems[END_REF].

We will rely on the following version of Theorem 5.5 from [START_REF] Billingsley | Convergence of probability measures[END_REF].

Theorem 3.1. Let X n d → X, with X n and X assuming values in a separable metric space S. Let ϕ n , ϕ : S → S be measurable functions such that for any x ∈ S and any sequence x n → x we have

ϕ n (x n ) → ϕ(x). Then ϕ n (X n ) d → ϕ(X).
Note that except the equality K 0 (a, b, c) = Gamma(a, c), in case b > a > 0 we have K 1 (a, b, 0) = Beta II (a, b -a), where the Beta II (p, q) law with p, q > 0 is defined by the density

f (x) ∝ x p-1 (1+x) p+q 1 (0,∞) .
Several other distributions can be obtained by taking weak limits of Kummer laws.

Proposition 3.2. If α → ∞ then (1) when a > b (9) K α (a, b, c) w → Gamma(a -b, c),
(2) when b > a

(10) K 1 (a, b, c/α) w → Beta II (a, b -a), (3) when b > 0 (11) K α (αb + a 1 , αb + a 2 , c) w → GIG(a 1 -a 2 , c, b). (4) when b > 0 (12) K α (a √ α + b, a √ α, c) w → Gamma(b, c).
(5) when b > 0

(13) K α (aα, aα + b, c/α) w → InvGamma(b, a),
where InvGamma(b, c) is defined by the density

f (x) ∝ x -b-1 e -c/x 1 (0,∞) (x).
3.1. The Lukacs property. The following independence property is related to the characterization of the gamma laws proved in [START_REF] Lukacs | A characterization of the gamma distribution[END_REF].

Theorem 3.3. Assume that (X 1 , Y 1 ) ∼ Gamma(a 1 , c) ⊗ Gamma(b 1 , c). Let (U 1 , V 1 ) = Y1 X1 , X 1 + Y 1 . Then (U 1 , V 1 ) ∼ Beta II (b 1 , a 1 ) ⊗ Gamma(a 1 + b 1 , c).
Proof. In view of Theorem 2.1 with α = n, β = 0 and Remark 2.1 we see that

(X (n) 1 , Y 1 ) ∼ K n (a 1 + b 1 , b 1 , c) ⊗ Gamma(b 1 , c) implies that for ϕ n (x, y) = ny 1+nx , x 1+n(x+y) 1+nx
we have [START_REF] Koudou | Independence properties of the Matsumoto-Yor type[END_REF] ϕ n (X

(n) 1 , Y 1 ) ∼ K 1 (b 1 , a 1 + b 1 , c/n) ⊗ Gamma(a 1 + b 1 , c). Since, see (9), K n (a 1 + b 1 , b 1 , c) w → Gamma(a 1 , c) as n → ∞, we see that (15) (X (n) 1 , Y 1 ) d → (X 1 , Y 1 ) ∼ Gamma(a 1 , c) ⊗ Gamma(b 1 , c).
Moreover, for x n → x > 0 and y > 0

ϕ n (x n , y) = y xn+1/n , xn xn+1/n (x n + y + 1/n) → y x ,
x + y =: ϕ(x, y). Thus, in view of (15), Theorem 3.1 implies [START_REF] Letac | About an extension of the Matsumoto-Yor property[END_REF] ϕ n (X

(n) 1 , Y 1 ) d → ϕ(X 1 , Y 1 ) = Y1 X1 , X 1 + Y 1 .
On the other hand, [START_REF] Koudou | Independence properties of the Matsumoto-Yor type[END_REF], in view of (10), yields [START_REF] Lukacs | A characterization of the gamma distribution[END_REF] P ϕn(X (n)

1 ,Y1) w → Beta II (b 1 , a 1 ) ⊗ Gamma(a 1 + b 1 , c).
Comparing ( 16) and ( 17) we conclude that

(U 1 , V 1 ) = Y1 X1 , X 1 + Y 1 ∼ Beta II (b 1 , a 1 ) ⊗ Gamma(a 1 + b 1 , c). □ 3.2.
The Kummer-Gamma independence property. The following result was proved in [START_REF] Koudou | Independence properties of the Matsumoto-Yor type[END_REF]; see also [START_REF] Weso Lowski | On the Matsumoto-Yor type regression characterization of the gamma and Kummer distributions[END_REF].

Theorem 3.4. Assume that (X 2 , Y 2 ) ∼ K(a 2 , a 2 + b 2 , c 2 ) ⊗ Gamma(b 2 , c 2 ). Let (U 2 , V 2 ) = X 2 + Y 2 , 1+ 1 X2+Y2 1+ 1 X2
.

Then (U 2 , V 2 ) ∼ K(a 2 + b 2 , a 2 , c 2 ) ⊗ Beta I (a 2 , b 2 )
, where Beta I (p, q) has the density

f (y) ∝ y p-1 (1 -y) q-1 1 (0,1) (y).
Proof. In view of Theorem 2.1 with α = 1 and β = n we see that

(X 2 , Y (n) 2 ) ∼ K 1 (a 2 , a 2 + b 2 , c 2 ) ⊗ K n (a 2 + b 2 , a 2 , c 2 ) implies that for ϕ n (x, y) = y 1+n(x+y) 1+x+ny , nx 1+x+y 1+x+ny we have (18) ϕ n (X 2 , Y (n) 2 ) ∼ K 1 (a 2 + b 2 , a 2 , c 2 ) ⊗ K 1 (a 2 , a 2 + b 2 , c 2 /n). Since, see (9), K n (a 2 + b 2 , a 2 , c) w → Gamma(b 2 , c 2 ), we see that (19) P X2, Y (n) 2 w → K 1 (a 2 , a 2 + b 2 , c 2 ) ⊗ Gamma(b 2 , c 2 ).
Moreover, for any x > 0 and y n → y > 0 we have

ϕ n (x, y n ) = yn yn+(1+x)/n (x + y n + 1 n ),
x yn+(1+x)/n (1 + x + y n ) → x + y, x y (1 + x + y) =: ϕ(x, y). Consequently, by Theorem 3.1, in view of [START_REF] Matsumoto | Interpretation via Brownian motion of some independence properties between GIG and gamma laws[END_REF], we have [START_REF] Piliszek | Regression conditions that characterize free-Poisson and free-Kummer distributions[END_REF] ϕ n (X 2 , Y

(n) 2 ) d → ϕ(X 2 , Y 2 ) = X 2 + Y 2 , X2 Y2 (1 + X 2 + Y 2 )
. On the other hand, [START_REF] Matsumoto | An analogue of Pitman's 2M -X theorem for exponential Wiener functionals. Part II: the role of the generalized inverse Gaussian laws[END_REF], in view of (10), yields [START_REF] Piliszek | Kummer and gamma laws through independencies on trees -another parallel with the Matsumoto-Yor property[END_REF] P ϕn(X2,Y (n)

2

) w → K 1 (a 2 + b 2 , a 2 , c) ⊗ Beta II (a 2 , b 2 ).
Comparing [START_REF] Piliszek | Regression conditions that characterize free-Poisson and free-Kummer distributions[END_REF] and ( 21) we conclude that

(U 2 , V 2 ) := X 2 + Y 2 , X2 Y2 (1 + X 2 + Y 2 ) ∼ K 1 (a 2 + b 2 , a 2 , c) ⊗ Beta II (a 2 , b 2 ). But for h(u, v) = (u, v 1+v ) and ϕ = h • ϕ we have ϕ(x, y) = x + y, x y (1+x+y) 1+ x y (1+x+y) = x + y, 1+ 1 x+y 1+ 1 x . Thus (U 2 , V 2 ) = X 2 + Y 2 , 1+ 1 X2+Y2 1+ 1 X2 = ϕ(X 2 , Y 2 ) = U 2 , V2 1+ V2 ∼ K 1 (a 2 + b 2 , a 2 , c) ⊗ Beta I (a 2 , b 2 ).
Here we used the fact that if Z ∼ Beta II (p, q) then Z 1+Z ∼ Beta I (p, q). □ 3.3. The Matsumoto-Yor property. The following result was proved in Matsumoto and Yor [START_REF] Matsumoto | An analogue of Pitman's 2M -X theorem for exponential Wiener functionals. Part II: the role of the generalized inverse Gaussian laws[END_REF][START_REF] Matsumoto | Interpretation via Brownian motion of some independence properties between GIG and gamma laws[END_REF]; see also [START_REF] Letac | An independence property for the GIG and gamma laws[END_REF].

Theorem 3.5. Assume that

(X 3 , Y 3 ) ∼ GIG(-a 3 , b 3 , c 3 ) ⊗ Gamma(a 3 , b 3 ). Let (U 3 , V 3 ) = 1 X3+Y3 , 1 X3 -1 X3+Y3 . Then (U 3 , V 3 ) ∼ GIG(-a 3 , c 3 , b 3 ) ⊗ Gamma(a 3 , c 3 ).
Proof. In view of Theorem 2.1 with α = n and β = n 2 we see that if

(22) X (n) 3 , Y (n) 3 ∼ K n (nc 3 , nc 3 + a 3 , b 3 ) ⊗ K n 2 (nc 3 + a 3 , nc 3 , b 3 ) then with ϕ n (x, y) = y 1+n 2 (x+y) 1+nx+n 2 y , nx 1+n(x+y) 1+nx+n 2 y we have (23) ϕ n X (n) 3 , Y (n) 3 ∼ K n (nc 3 + a 3 , nc 3 , b 3 ) ⊗ K n (nc 3 , nc 3 + a 3 , b 3 /n).
Since, see ( 11) and ( 12),

K n (nc 3 , nc 3 + a 3 , b 3 ) w → GIG(-a 3 , b 3 , c 3 ) and K n 2 (nc 3 + a 3 , nc 3 , b 3 ) w → Gamma(a 3 , b 3 )
we see that ( 24)

X (n) 3 , Y (n) 3 d → (X 3 , Y 3 ) ∼ GIG(-a 3 , b 3 , c 3 ) ⊗ Gamma(a 3 , b 3 ).
Moreover, if x n → x > 0 and y n → y > 0 then we have

ϕ n (x n , y n ) = y y+ x n + 1 n 2 (x + y + 1 n 2 ), x x+y+ 1 n y+ x n + 1 n 2 → x + y, x (x+y) y .
Consequently, by [START_REF] Sasada | Yang-Baxter maps and independence preserving property[END_REF], in view of Theorem 3.1 we have (25)

ϕ n X (n) 3 , Y (n) 3 d → ϕ(X 3 , Y 3 ) = X 3 + Y 3 , X3(X3+Y3)

Y3

.

On the other hand, since ( 11) and ( 13),

K n (nc 3 + a 3 , nc 3 , b 3 ) w → GIG(a 3 , b 3 , c 3 ) and K n (nc 3 , nc 3 + a 3 , b 3 /n) w → InvGamma(a 3 , c 3 )
we conclude that (26)

P ϕn X (n) 3 , Y (n) 3 w → GIG(a 3 , b 3 , c 3 ) ⊗ InvGamma(a 3 , c 3 ).
Combining [START_REF] Seppäläinen | Scaling for a one-dimensional directed polymer with boundary conditions[END_REF] with [START_REF] Weso Lowski | On the Matsumoto-Yor type regression characterization of the gamma and Kummer distributions[END_REF] we get

( U 3 , V 3 ) := X 3 + Y 3 , X3(X3+Y3) Y3 ∼ GIG(a 3 , b 3 , c 3 ) ⊗ InvGamma(a 3 , c 3 ). Denoting h(u, v) = (u -1 , v -1 ) and ϕ = h • ϕ, we get ϕ(x, y) = 1 x+y , y x(x+y) = 1 x+y , 1 x -1 x+y . Finally, we obtain (U 3 , V 3 ) = 1 X3+Y3 , 1 X3 -1 X3+Y3 = ϕ(X 3 , Y 3 ) = 1 U3 , 1 V3 ∼ GIG(-a 3 , c 3 , b 3 ) ⊗ Gamma(a 3 , c 3 ) □ 3.4.
The discrete KdV independence property. The following result was proved in [START_REF] Croydon | Detailed balance and invariant mesures for discrete KdV-and Toda-type systems[END_REF]; see also [START_REF] Bao | Characterizations of the generalized inverse Gaussian, asymmetric Laplace, and shifted (truncated) exponential laws via independence properties[END_REF] and [START_REF] Letac | About an extension of the Matsumoto-Yor property[END_REF].

Theorem 3.6. Assume that

(27) (X 4 , Y 4 ) ∼ GIG(-a 4 , αb 4 , c 4 ) ⊗ GIG(-a 4 , βc 4 , b 4 ). Let (U 4 , V 4 ) = Y 4 1+αX4Y4 1+βX4Y4 , X 4 1+βX4Y4 1+αX4Y4 . Then (28) (U 4 , V 4 ) ∼ GIG(-a 4 , αc 4 , b 4 ) ⊗ GIG(-a 4 , βb 4 , c 4 ).
Proof. In view of Theorem 2.1 with α changed into n/α and β changed into n/β we see that if

(29) X (n) 4 , Y (n) 4 ∼
ϕ n X (n) 4 , Y (n) 4 d → ϕ( X 4 , Y 4 ) = α Y 4 X4+ Y4 β X4+α Y4 , β X 4 X4+ Y4
β X4+α Y4 . On the other hand since, see again [START_REF] Ferguson | A characterization of the exponential distribution[END_REF], Note that for g(x, y) = (1/x, βy), in view of ( 27) and (31), we have

K n/α (
g(X 4 , Y 4 ) d = ( X 4 , Y 4 ) (note that if R ∼ GIG(p, a, b) anf if k > 0, then kR ∼ GIG(p, a/k, kb)
) and for h(x, y) = (x/α, 1/y), in view of ( 32) and (28), we see that

h( U 4 , V 4 ) d = (U 4 , V 4 ). Therefore, defining ϕ = h • ϕ • g we get (U 4 , V 4 ) d = ϕ(X 4 , Y 4 ). But ϕ(x, y) = y 1+αxy 1+βxy , x 1+βxy 1+αxy ,
which concludes the proof. □

The Kummer transform and the characterization

For a positive random variable W and γ ≥ 0 consider an extended Laplace transform L

W of the form

L (γ) W (s, t, z) = E W s (1+γW
) t e -zW . We will call it the Kummer transform. Note that the Kummer tranform is well defined at least for s, z > 0 and t ∈ R. Moreover, for any fixed s > 0, t ∈ R, the Kummer transform as a function of z > 0, is just the Laplace transform of the measure w s (1+γw) t P W (dw), so it uniquely determines the distribution of W . Note also that ( 35)

L (γ) W (s, t, z) + γL (γ) W (s + 1, t, z) = L (γ)
W (s, t -1, z) and for any k = 1, 2, . . .

(36) ∂ k L (γ) W (s,t,z) ∂z k = (-1) k L (γ) W (s + k, t, z). Proposition 4.1. Let X ∼ K α (a, b, c), a, c > 0, b ∈ R. Then (37) L (α) X (s, t, z) = Γ(a+s)U a+s,a+s-b-t+1, c+z α α s Γ(a)U (a,a-b+1, c α ) , s > 0, t ∈ R, z > -c
, where U is the Kummer function (see 13.2.5 in [START_REF] Abramowitz | Pocketbook of Mathematical functions[END_REF]) defined by

(38) U (a, b, z) = 1 Γ(a) ∞ 0 x a-1 (1+x) a-b+1 e -zx dx, a, z > 0, b ∈ R.
U (a, a + 1, z) = z -a ,
whence, if X ∼ K α (a, 0, c) then (37) gives

L (α)
X (0, 0, z) = c a (c+z) a , which implies that X is a Gamma random variable, Gamma(a, c).

Proposition 4.2. Let b ∈ R, a, c, α > 0. Assume that for some fixed (s, t) ∈ (0, ∞) × R and all z > 0 (39) L (α) X (s, t, z) = k(s, t)U a + s, a + s -b -t + 1, c+z α ,
where k(s, t) is a constant (depending also on α, a, b, c). Then X ∼ K α (a, b, c).

Proof. Fix some z 0 > 0. Then (39) implies ( 40)

L (α) X (s,t,z+z0) L (α) X (s,t,z0) = U a+s,a+s-b-t+1, c+z0+z α U a+s,a+s-b-t+1, c+z0 α , z ≥ 0.
This is the Laplace transform of a random variable Y with distribution (41) P Y (dy) = y s (1+αy) t e -z 0 y P X (dy) L ] Let X and Y be independent and U, V are defined as in [START_REF] Billingsley | Convergence of probability measures[END_REF]. In view of (5), ( 6) and (7) we then see that if

(42) L (α) X (s, t, z) L (β) Y (t, s, z) = L (α) U (t, s, z) L (β)
V (s, t, z), (s, t, z) ∈ (0, ∞) × R × (0, ∞) then U and V are independent.

Let (X, Y ) ∼ K α (a, b; c) ⊗ K β (b, a; c). Consequently, to prove Theorem 2.1, it suffices to check that identity (42) is satisfied for (U, V ) ∼ K α (b, a; c) ⊗ K β (a, b; c). Referring to (37), we see that (42) is then equivalent to

(43) U a+s,a+s-b-t+1, c+z α U b+t,b+t-a-s+1, c+z β α s β t U (a,a-b+1, c α ) U b,b-a+1, c β = U b+t,b+t-a-s+1, c+z α U a+s,a+s-b-t+1, c+z β α t β s U (b,b-a+1, c α ) U a,a-b+1, c β .
To see that (43) holds true we rely on the following identity for the Kummer function U (see (13.1.29) in [START_REF] Abramowitz | Pocketbook of Mathematical functions[END_REF])

(44) U (a, b, z) = z 1-b U (1 + a -b, 2 -b, z).
Note also, that in view of (42) and (37) of Proposition 4.1 we have

L (α) X (s,t,z) L (α) U (t,s,z) = L (β) V (s,t,z) L (β) Y (t,s,z) = c a-b Γ(b) Γ(a) Γ(a+s) Γ(b+t) (c + z) b-a+t-s .
Now we are ready to formulate the main result which is a characterization of Kummer laws by the detailed balance condition with respect to the function F given in (3). 1 and symmetry with respect to α and β we conclude that the cases β = 0, α > 0 and β > 0, α = 0 also follow immediately from Theorem 2.6. of [START_REF] Piliszek | Change of measure technique in characterizations of the Kummer and gamma laws[END_REF]. That is, we need only to prove Theorem 4.3 for α > 0 and β > 0. The proof of this case requires much more efforts and is given in several steps in the next section.

5.

Proof of Theorem 4.3 for α > 0 and β > 0 5.1. Independence through Kummer transforms. We first note that the assumed independence properties imply equality (42) for (s, t, z) ∈ (0, ∞) × R × (0, ∞). Differentiating (42) with respect to z and dividing side-wise resulting equality by (42) we get

.

Using identity (35) we obtain

β L (α) X (s,t-1,z) L (α) X (s,t,z) + α L (β) Y (t,s-1,z) L (β) Y (t,s,z) = β L (α) U (t,s-1,z) L (α) U (t,s,z) + α L (β) V (s,t-1,z) L (β) Y (s,t,z)
.

Changing in the above formula s to s + 1 and t to t + 1 we arrive at

(46) β L (α) X (s+1,t,z) L (α) X (s+1,t+1,z) + α L (β) Y (t+1,s,z) L (β) Y (t+1,s+1,z) = β L (α) U (t+1,s,z) L (α) U (t+1,s+1,z) + α L (β) V (s+1,t,z) L (β) Y (s+1,t+1,z)
. Subtracting side-wise (45) (multiplied by αβ) from (46), in view of (35), we get

(47) β M (α) X (s, t, z) + α M (β) Y (t, s, z) = β M (α) U (t, s, z) + α M (β) V (s, t, z), where M (γ) W (s, t, z) = L (γ) W (s+1,t,z)L (γ) W (s,t+1,z) L (γ) W (s,t,z)L (γ) W (s+1,t+1,z)
.

From now on we suppress superscripts in L and M functions, that is we write

M X := M (α) X , L X := L (α) X , M Y := M (β) Y , L Y := L (β) Y , M U := M (α) U , L U := L (α) U and M V := M (β) V , L V := L (β) V . Note that (42) implies (48) M X (s, t, z)M Y (t, s, z) = M U (t, s, z)M V (s, t, z).
Combining (47) with (48) we get

(β M X (s, t, z) -α M V (s, t, z)) (M X (s, t, z) -M U (t, s, z)) = 0 (49) (β M U (t, s, z) -α M Y (t, s, z)) (M V (s, t, z) -M Y (t, s, z)) = 0. (50) Since M X , M Y , M U , M V all extend uniquely to meromorphic functions in a common domain in C 3 it follows from (49) that either β M X ≡ α M V or M X ≡ M U and from (50) that either β M U ≡ α M Y or M V ≡ M Y .
In Section 5.2 we will prove that β M X ≡ α M V is impossible. It will follow by symmetry that also β M U ≡ α M Y is impossible. Then, in Section 5.3 we will consider the case M X ≡ M U . It relates L X and L U through auxiliary functions a, b, f which are deciphered in Section 5.4. In Section 5.5 we eliminate L X from the problem. It leads to the Kummer differential equation for slightly modified version of L U , which allows to identify L U (and thus also L X ), up to parameters. The case M V ≡ M Y follow by the analogous approach. The final step in Section 5.6 lies in identification of relations between parameters of all four transforms: L X , L Y , L U and L V .

The case βM

X ≡ αM V is impossible. Assume (51) βM X (s, t, z) = αM V (s, t, z), s, t ≥ 0, z > 0. Define (52) A(s, t, z) := L X (s+1,t)L V (s,t) L X (s,t)L V (s+1,t)
and (53) B(s, t, z) := L X (s,t)L V (s,t+1) L X (s,t+1)L V (s,t) , where we suppressed the argument z. Note that (51) implies

A(s, t + 1, z) = β α A(s, t, z) and B(s + 1, t, z) = α β B(s, t, z), s, t ∈ N. Consequently, A(s, t, z) = β α t a(s, z), and B(s, t, z) = α β s b(t, z),
where a(s, z) = A(s, 0, z) and b(t, z) = B(0, t, z).

Note that (51) implies also (54) β A(s, t, z) L X (s,t) L X (s+1,t+1) = α B(s, t, z) L V (s,t) L V (s+1,t+1) . Consequently, h(s, t, z) := a(s,z) b(t,z) = α β t+s+1 L X (s+1,t+1)L V (s,t) L X (s,t)L V (s+1,t+1) . Then

β α s+t+1 ∂h(s,t,z) ∂z = Num [L X (s,t)L V (s+1,t+1)] 2 , where the numerator Num = [L X (s + 2, t + 1)L V (s, t) + L X (s + 1, t + 1)L V (s + 1, t)] L X (s, t)L V (s + 1, t + 1) -L X (s + 1, t + 1)L V (s, t) [L X (s + 1, t)L V (s + 1, t + 1) + L X (s, t)L V (s + 2, t + 1)] =L X (s, t)L V (s, t) [L X (s + 2, t + 1)L V (s + 1, t + 1) -L X (s + 1, t + 1)L V (s + 2, t + 1)] + L X (s, t)L X (s + 1, t + 1)L V (s + 1, t)L V (s + 1, t + 1) -L X (s + 1, t)L X (s + 1, t + 1)L V (s, t)L V (s + 1, t + 1) = I 1 + I 2 -I 3 .
Note that the last two summands in the above expression can be rewritten with the help of (35) as follows:

I 2 = 1 β L X (s, t)L X (s + 1, t + 1)L V (s + 1, t) [L V (s, t) -L V (s, t + 1)] and I 3 = 1 α L X (s + 1, t) [L X (s, t) -L X (s, t + 1)] L V (s, t)L V (s + 1, t + 1).
Consequently,

I 1 + I 2 -I 3 = L X (s, t)L V (s, t) 1 α L V (s + 1, t + 1) [αL X (s + 2, t + 1) -L X (s + 1, t)] -1 β L X (s + 1, t + 1) [βL V (s + 2, t + 1) -L V (s + 1, t)] + 1 α L X (s + 1, t)L X (s, t + 1)L V (s, t)L V (s + 1, t + 1) -1
β L X (s, t)L X (s + 1, t + 1)L V (s + 1, t)L V (s, t + 1). Note that (51) implies that the two last terms cancel. Therefore referring again to (35) in the first two expressions above we get

Num = L X (s, t)L V (s, t)L X (s + 1, t + 1)L V (s + 1, t + 1) α-β αβ whence ∂ h(s,t,z) ∂ z = h(s, t, z) α-β αβ , i.e.
h(s, t, z) = χ(s, t) exp α-β αβ z , where χ(s, t) does not depend on z. Consequently,

a(s,z) b(t,z) = a(s) b(t) e z α-β αβ ,
where a(s) = a(s, 0) and b(t) = b(0, t).

Therefore, referring to (54), for s = t = 0 we obtain

e - 1 β - 1 α z (0,∞) 2 x 1+αx e -z(x+v) P X (dx)P V (dv) = β a(0) α b(0) (0,∞) 2 v 1+βv e -z(x+v) P X (dx)P V (dv)
which, with η = 1 β -1 α , can be written as (55)

R×(0,∞) 2

x 1+αx e -z(w+x+v) δ η (dw)P X (dx)P V (dv) = β a(0) α b(0) (0,∞) 2 v 1+βv e -z(x+v) P X (dx)P V (dv).

In view of (55) we see that the support of the convolution δ η * P X * P V (determined by the left-hand side of (55)) coincides with the support of the convolution P X * P V (determined by the right-hand side of (55)). Consequently, denoting the latter support by W and letting w 0 = inf W, we get η + w 0 = w 0 , which is impossible since by assumption α ̸ = β.

5.3.

The case of M X ≡ M U and functions a, b and f . We consider the equation (56) M X (s, t, z) = M U (t, s, z), s, t ∈ {0, 1, . . .}, z > 0.

Denote (57)

A(s, t, z) := L X (s+1,t)L U (t,s) L X (s,t)L U (t,s+1) , and (58) B(t, s, z) := L U (t+1,s)L X (s,t) L U (t,s)L X (s,t+1) , where we skipped the superscript (α) and the argument z in L X and L U .

Note that (56) implies that for all s, t ∈ N = {0, 1, . . .} we have A(s, t, z) = A(s, t + 1, z) and B(t, s, z) = B(t, s + 1, z).

Consequently, for (s, t) ∈ N 2 we have A(s, t, z) = A(s, 0, z) =: A(s, z) and B(t, s, z) = B(t, 0, z) =: B(t, z). Now (56) can be written as ,s+1) .

(59) A(s, z) L X (s,t) L X (s+1,t+1) = B(t, z) L U (t,s) L U (t+1

Consider now the quotient

A(s,z) B(t,z) = L X (s+1,t+1)L U (t,s) L X (s,t)L U (t+1,s+1) . Then the numerator Num of the derivative

∂ A(s,z) B(t,z) ∂z assumes the form Num =[L X (s + 2, t + 1)L U (t, s) + L X (s + 1, t + 1)L U (t + 1, s)]L X (s, t)L U (t + 1, s + 1) -L X (s + 1, t + 1)L U (t, s)[L X (s + 1, t)L U (t + 1, s + 1) + L X (s, t)L U (t + 2, s + 1)] =L X (s, t)L U (t, s)[L X (s + 2, t + 1)L U (t + 1, s + 1) -L X (s + 1, t + 1)L U (t + 2, s + 1)] +L X (s, t)L X (s + 1, t + 1)L U (t + 1, s) 1 α [L U (t, s) -L U (t, s + 1)] -L X (s + 1, t) 1 α [L X (s, t) -L X (s, t + 1)]L U (t, s)L U (t + 1, s + 1)
, where we twice used (35). Referring again to (56), after cancellation, we get

α Num L X (s,t)L U (t,s) =L U (t + 1, s + 1) [αL X (s + 2, t + 1) -L X (s + 1, t)] -L X (s + 1, t + 1) [αL U (t + 2, s + 1) -L U (t + 1, s)] .
Note that (35) applied to the expressions in square brackets above gives -L X (s + 1, t + 1) for the first square bracket and -L U (t + 1, s + 1) for the second. Consequently, Num = 0 and thus A(s,z) B(t,z) = a(s) b(t) , where a(s) := A(s, 0) and b(t) := B(t, 0). Consequently, we have the representations:

(60) A(s, z) = f (z)a(s) and B(t, z) = f (z)b(t), z > 0, s, t ∈ N,
where f = A(0,z) a(0) = B(0,z) b(0) . Note that (59) can be rewritten as (61) a(s) L X (s,t,z) L X (s+1,t+1,z) = b(t) L U (t,s,z) L U (t+1,s+1,z) . 5.4. Computing a, b and f . Taking logarithms of (57) sidewise, using the first equality of (60) and differentiating with respect to z, in view of (36), we obtain

f ′ f = -L X (s+2,t) L X (s+1,t) + L X (s+1,t) L X (s,t) -L U (t+1,s) L U (t,s) + L U (t+1,s+1) L U (t,s+1) . Note that ∂ 2 ∂z 2 log L X (s, t) = L X (s+2,t) L X (s,t) -L X (s+1,t) L X (s,t) 2 .
Using the above formula, the identity (36) and recalling the definition of M U we finally get

f ′ f ∂ ∂z log L X (s, t) = ∂ 2 ∂z 2 log L X (s, t) + L X (s+1,t)L U (t+1,s) L X (s,t)L U (t,s) (1 -M -1 U (t, s, z)).
Starting with (58), in a similar way, we obtain the analogue of the above

f ′ f ∂ ∂z log L U (t, s) = ∂ 2 ∂z 2 log L U (t, s) + L U (t+1,s)L X (s+1,t) L U (t,s)L X (s,t)
(1 -M -1 X (s, t, z)). Subtracting the last two equalities sidewise, in view of (56), we obtain

f ′ f = g ′ g , where g = ∂ ∂z log L U (t,s) L X (s,t) . Consequently, (62) K(s, t)f = ∂ ∂z log L U (t,s) L X (s,t) = L X (s+1,t) L X (s,t) -L U (t+1,s) L U (t,s)
for some function K which does not depend on z. Referring to (60) as well as to (57) and (58) again we get K(s, t) = a(s) L U (t,s+1) L U (t,s) -b(t) L X (s,t+1) L X (s,t) .

For L X (s, t) and L X (s + 1, t) in (67) insert the relevant expression from (64) and (65), respectively. Together with (66) it yields a(s)L U (t, s + 1) -K(s, t)L U (t, s) = (c + z)L U (t + 1, s).

Regarding L U (t, s) and L U (t + 1, s) as the right-hand side of identity (35) with suitable s and t we get

α(c + z)L U (t + 2, s + 1) + (c + z + αK(s, t))L U (t + 1, s + 1) -b(t)L U (t, s + 1) = 0.
In view of (36), the above equation transforms into the second order diferential equation for the function h := L U (t, s + 1) as follows

α(c + z) h ′′ (z) + α(b(t) -a(s)) -(c + z)) h ′ (z) -b(t) h(z) = 0.
Consequently, for g defined by g(z) = h(αz -c) we get the Kummer equation Returning to L U (t, s) (recall that g was defined through L U (t, s + 1)) we get L U (t, s, z) = c U (s, t)U (b + t, b + t -a -s + 1, c+z α ), with a, b > 0 and c ≥ 0.

Changing the roles of L X and L U in the above argument starting with (64) we obtain L X (s, t, z) = c X (s, t)U (a + s, a + s -b -t + 1, c+z α ). Assume that c = 0. Recalling (38) we see that: (1) if a ̸ = b then either U (a, a -b + 1, 0) = ∞ or U (b, b -a + 1, 0) = ∞; (2) if a = b then U (a, 1, 0) = U (b, 1, 0) = ∞. Since L X (0, 0, 0) = L U (0, 0, 0) = 1 we obtain thus a contradiction. Therefore c > 0 and Proposition 4.2 implies that X ∼ K α (a, b, c) and U ∼ K α (b, a, c).

In case κ = 0 we have f (z) = C ̸ = 0 and A(s, z) = a > 0 and B(t, z) = b > 0 where a = ãC and b = bC. We now show that this is impossible. Indeed, (62) then yields a -b = L X (s+1,t) L X (s,t) -L U (t+1,s) L U (t,s) . Combining this with (58) we get (a -b)L X (s, t) = L X (s + 1, t) -bL X (s, t + 1).

For s = t = 0 we thus get E((X -a + b)e -zX = bE 1 1+αX e -zX . Consequently, (x -a + b)P X (dx) = b 1+αx P X (dx). Equivalently, (1+αx)(x-a+b) b P X (dx) = P X (dx).

Since (1 + αx)(x -a + b) = b is equivalent to αx 2 + (α(b -a) + 1)x -a = 0, the roots of which have different signs. Since X is nonnegative this would imply that its support degenerates to a point, which contradicts our assumptions.

5.6. Identifying the parameters. We have proved that X ∼ K α (a, b, c), U ∼ K α (b, a, c), Y ∼ K β (ã, b, c) and V ∼ K β ( b, ã, c) for some a, b, c, ã, b, c > 0. Using (37) for each of the variables X, Y, U, V , equation (42) reads:

  and, in view of Remark 2.1, reference to Theorem 2.1 ends the proof. □

Proof.□Remark 4 . 1 .

 41 It is a simple consequence of that fact that due to the definition of the Kummer function U in (38) the normalizing constant of the Kummer distribution K α (a, b, c) has the form α a Γ(a)U (a,a-b+1, c α ). Note that when b = a + 1, in view of (38), we have

.

  In view of Proposition 4.1, by (37) and (40), the random variable Y has the Kummer distribution K α (a + s, b + t, c + z 0 ), i.e. P Y (dy) ∝ y a+s-1 (1+αy) b+t e -(c+z0)y dy. The result follows by comparing the last formula with (41). □ Remark 4.2. [Alternative proof of Theorem 2.1.

Theorem 4 . 3 .Remark 4 . 3 .

 4343 Let α, β ≥ 0, α ̸ = β. Let X, Y be positive, independent, non-Dirac random variables and define U = Y 1+β(X+Y ) 1+αX+βY and V = X 1+α(X+Y ) 1+αX+βY . If U and V are independent, then there exist a, b, c > 0 such that(X, Y ) ∼ K α (a, b; c) ⊗ K β (b, a; c).We then have(U, V ) ∼ K α (b, a; c) ⊗ K β (a, b; c).Recall that Theorem 2.6 of[START_REF] Piliszek | Change of measure technique in characterizations of the Kummer and gamma laws[END_REF] says that for random variables X and Y which are independent, positive, non-Dirac, independence of U and V given by (1) implies that X ∼ K(a, b, c) and Y ∼ Gamma(b, c). (Note the change of parametrization of the Kummer distribution: instead of b -a in[START_REF] Piliszek | Change of measure technique in characterizations of the Kummer and gamma laws[END_REF] we write here just b.) In view of the first part of Remark 2.1 this result covers the case (α, β) = (1, 0) in Theorem 4.3. Due to the second part of Remark 2.

  (68)zg ′′ (z) + (b(t) -a(s) -z)g ′ (z) -b(t)g(z) = 0.It is well known that the general solution is of the formg(z) = c 1 M (b(t), b(t) -a(s), z) + c 2 U (b(t), b(t) -a(s), z),see 13.1.1, 13.1.2 and 13.1.3 in[START_REF] Abramowitz | Pocketbook of Mathematical functions[END_REF]). Recall that M (a, b, z) is unbounded when z → ∞ (see e.g. 13.1.4 in[START_REF] Abramowitz | Pocketbook of Mathematical functions[END_REF]) and U (a, b, z) → 0 as z → ∞. Since g, as a Laplace transform of a probability measure, is bounded, we necessarily have g(z) = c U (s, t)U (b(t), b(t) -a(s), z).

Γ

  (a+s)U a+s,a+s-b-t+1, c+z α α s Γ(a)U (a,a-b+1, c α ) × Γ(ã+t)U ã+t,ã+t-b-s+1, c+z β β t Γ(ã)U ã,ã-b+1, c β = Γ(b+t)U b+t,b+t-a-s+1, c+z α α t Γ(b)U (b,b-a+1, c α ) × Γ( b+s)U b+s, b+s-ã-t+1, c+z β β s Γ( b)U b, b-ã+1, c β which, by applying identity (44) to the left-hand side, gives )U b+t,b+t-a-s+1, c+z α α t Γ(b)U (b,b-a+1, c α ) × Γ( b+s)U b+s, b+s-ã-t+1, c+z β β s Γ( b)U b, b-ã+1, c β.After cancellations we obtain (69)(c+z) b+t-a-s (c+z) b+s-ã-t c b-a cb -ã × Γ(a+s) Γ(a) × Γ(ã+t) Γ(ã) = Γ(b+t) Γ(b) × Γ( b+s) Γ( b). Taking the logarithm and differentiating in z gives b+t-a-s c+z+ b+s-ã-t c+z = 0.Since this holds for any z > 0, we conclude that c = c and b -a = ã -b. Returning to (69) we haveΓ(a+s) Γ(a) × Γ(ã+t) Γ(ã) = Γ(b+t) Γ(b) × Γ( b+s) Γ( b), which for (s, t) = (0, 1), yields ã = b and for (s, t) = (1, 0) yields a = b.

  nb 4 , nb 4 + a 4 , c 4 ) , c 4 , αb 4 ) ⊗ GIG(a 4 , c 4 , βb 4 ). , V 4 ) := ϕ( X 4 , Y 4 ) ∼ GIG(-a 4 , c 4 , αb 4 ) ⊗ GIG(a 4 , c 4 , βb 4 ).

	by (30) we conclude that
	(33) → GIG(-a 4 Combining (32) and (33) we get P ϕn(X (n) 4 ,Y (n) 4 ) w
	(34)	( U 4
		w → GIG(-a 4 , c 4 , αb 4 )
	and
		K n/β (nb 4 + a 4 , nb 4 , c 4 )	w → GIG(a 4 , c 4 , βb 4 )

Acknowledgement: J. Weso lowski was supported by grant Beyond POB II no. 1820/366/Z01/2021 within the Excellence Initiative: Research University (IDUB) programme of the Warsaw Univ. of Technology, Poland.

Now, (35) applied to L U (t, s + 1) and L X (s, t + 1) gives K(s, t) = a(s) -b(t) -α a(s) L U (t+1,s+1) L U (t,s) -b(t) L X (s+1,t+1) L X (s,t)

.

Referring to (61) we see that the expression in parenthesis above is zero, whence K(s, t) = a(s) -b(t). Now we write (62) for s + 1 and t, which gives s+1,t) . Substracting sidewise this equality from (62) we get

where the last equality follows from the first part of (60).

Similarly, using (62) with s and t + 1 we get

dz , where the last equality follows from the second part of (60).

Consequently a(s) -a(s

In case κ ̸ = 0 we have A(s, z) = a+s c+z and B(t, z) = b+t c+z , where a := ã/κ, b := b/κ and c = C/κ. Since A(s, z) and B(t, z) are strictly positive for all s ≥ 0, t ≥ 0 and z > 0, we conclude that a, b > 0 and c ≥ 0. Note that κ = 0 implies f ≡ const, which in view of (60) would imply A(s, z) = const and B(t, z) = const . 5.5. The Kummer ode and identification (up to parameters) of L X , L U , L Y and L V . Note that (62), when κ ̸ = 0, in view of (63), can be rewritten as t) , z > 0, where we changed K/κ into K and c = C/κ. (Note that with such a new K(s, t) we also have a(s) = s+a and b(t) = t + b.) Consequently, (z + c) K(s,t) = c(s, t) L U (t,s) L X (s,t) , where c(s, t) does not depend on z.

Rewrite the above as

and for s + 1 and t as

65) by (64) and referring to (57) we get (66) c(s+1,t) c(s,t) = L X (s+1,t)L U (t,s) L X (s,t)L U (t,s+1) (z + c) = a(s). Now we differentiate (64) with respect to z and get (67) K(s, t)(z + c) K(s,t)-1 L X (s, t) -(z + c) K(s,t) L X (s + 1, t) = -c(s, t)L U (t + 1, s).


Jovanna Arndt

Robert M Healy
email: robert.healy@ontario.ca

Ari Setyan

Pascal Flament

Karine Deboudt

Véronique Riffault

Laurent Y Alleman

Saliou Mbengue

John C Wenger

Characterization and source apportionment of single particles from metalworking activities

Keywords: Investigation, Methodology, Formal Analysis, Visualization, Data curation, Writing-original draft preparation Investigation, Methodology, Formal Analysis, Visualization Project Administration, Funding Acquisition. Karine Deboudt: Investigation, Conceptualization, Writing-review Investigation, Formal Analysis, Conceptualization, Writing-review and editing, Supervision, Project Administration, Particulate matter, source apportionment, mass spectrometry, industrial emissions, metalworking, air quality

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Negative health outcomes, lower crop yields, building damage and other impacts associated with industrial air pollution and greenhouse gas emissions were estimated to cost Europe over €300 billion between 2008and 2012(European Environment Agency, 2014). Almost a third of emissions of particulate matter with aerodynamic diameters smaller than 10 µm (PM 10 ) in France [START_REF] Citepa | Inventaire des émissions de polluants atmosphériques et de gaz à effet de serre en France -Format SECTEN[END_REF] are attributed to industrial sources. The city of Dunkirk lies in the Hauts de

France region of Northern France and is home to over 200 industrial facilities from many sectors including refining and cement production. Dunkirk also contains one of Europe's largest steelworks, which emits approximately 3,000 t of PM per year [START_REF] Flament | Iron isotopic fractionation in industrial emissions and urban aerosols[END_REF]. Numerous local industrial facilities generate plumes that impact air quality in the surrounding areas depending on local meteorological conditions, and local PM 10 concentrations regularly exceed the European threshold value (24-hour average of 50 µg m -3 ) [START_REF] Setyan | Investigation on the near-field evolution of industrial plumes from metalworking activities[END_REF].

The physico-chemical nature of industrial PM varies considerably, particularly in terms of composition, depending on the type of industrial processes involved [START_REF] Sanderson | A review of chemical and physical characterisation of atmospheric metallic nanoparticles[END_REF][START_REF] Taiwo | A review of receptor modelling of industrially emitted particulate matter[END_REF][START_REF] Riffault | Fine and Ultrafine Particles in the Vicinity of Industrial Activities: A Review[END_REF]. Ambient concentrations of particulate phase metals tend to be greater at locations directly influenced by industrial activities than at rural or urban locations (Osornio-Vargas [START_REF] Alvaro | Proinflammatory and cytotoxic effects of Mexico City air pollution particulate matter in vitro are dependent on particle size and composition[END_REF][START_REF] Schaumann | Metal-rich Ambient Particles (Particulate Matter2.5) Cause Airway Inflammation in Healthy Subjects[END_REF][START_REF] Lim | Airborne PM10 and metals from multifarious sources in an industrial complex area[END_REF][START_REF] Mohiuddin | Characterisation of trace metals in atmospheric particles in the vicinity of iron and steelmaking industries in Australia[END_REF]. Heavy metals and polycyclic aromatic hydrocarbons (PAHs) are important contributors to particle toxicity [START_REF] Harrison | Particulate matter in the atmosphere: which particle properties are important for its effects on health?[END_REF]. The generation of reactive oxygen species (ROS) and the development of asthma and cardiopulmonary disease have been linked to exposure to metals [START_REF] Schaumann | Metal-rich Ambient Particles (Particulate Matter2.5) Cause Airway Inflammation in Healthy Subjects[END_REF][START_REF] Majestic | Development of a Manganese Speciation Method for Atmospheric Aerosols in Biologically and Environmentally Relevant Fluids[END_REF]. PM 2.5 collected from the same industrial area of Dunkirk investigated in this work has been recently demonstrated to damage DNA through an in vitro assessment of immortalized lung cells [START_REF] Platel | Study of in vitro and in vivo genotoxic effects of air pollution fine (PM2.5-0.18) and quasi-ultrafine (PM0.18) particles on lung models[END_REF]. Thus, the impact of industrial emissions on downwind air quality needs to be assessed in order to better understand human exposure to the associated air toxics.

Particulate PAH and metal content studies have been performed in the Dunkirk area previosuly due to the importance of metalworking sources and their proximity to large residential areas [START_REF] Ledoux | Characterization of iron and manganese species in atmospheric aerosols from anthropogenic sources[END_REF][START_REF] Ledoux | Atmospheric aerosols behaviour at an industrial area in Northern France[END_REF][START_REF] Flament | Iron isotopic fractionation in industrial emissions and urban aerosols[END_REF][START_REF] Alleman | PM10 metal concentrations and source identification using positive matrix factorization and wind sectoring in a French industrial zone[END_REF][START_REF] Marris | Fast changes in chemical composition and size distribution of fine particles during the near-field transport of industrial plumes[END_REF][START_REF] Marris | Fe and Mn Oxidation States by TEM-EELS in Fine-Particle Emissions from a Fe-Mn Alloy Making Plant[END_REF][START_REF] Hleis | Chemical profile identification of fugitive and confined particle emissions from an integrated iron and steelmaking plant[END_REF][START_REF] Mbengue | Size-distributed metallic elements in submicronic and ultrafine atmospheric particles from urban and industrial areas in northern France[END_REF][START_REF] Cazier | Characterisation and seasonal variations of particles in the atmosphere of rural, urban and industrial areas: Organic compounds[END_REF][START_REF] Crenn | Fine particles sampled at an urban background site and an industrialized coastal site in Northern France-Part 2: Comparison of offline and online analyses for carbonaceous aerosols[END_REF]. In terms of source apportionment, the tracers used have included elemental ratios,

J o u r n a l P r e -p r o o f

for examples Mn/Fe as a signature for a ferromanganese alloy metalworking facility, V/Ni for petrochemical facilities and isotopes of Pb and Fe as a signature for a local steelworks [START_REF] Veron | Isotopic evidence of pollutant lead sources in Northwestern France[END_REF][START_REF] Ledoux | Characterization of iron and manganese species in atmospheric aerosols from anthropogenic sources[END_REF][START_REF] Flament | Iron isotopic fractionation in industrial emissions and urban aerosols[END_REF][START_REF] Ledoux | Atmospheric aerosols behaviour at an industrial area in Northern France[END_REF][START_REF] Alleman | PM10 metal concentrations and source identification using positive matrix factorization and wind sectoring in a French industrial zone[END_REF][START_REF] Marris | Fast changes in chemical composition and size distribution of fine particles during the near-field transport of industrial plumes[END_REF]. Emissions from the steelworks are also the dominant source of air emissions of Fe in Dunkirk, and the facility is estimated to emit 620 t of Fe annually [START_REF] Flament | Iron isotopic fractionation in industrial emissions and urban aerosols[END_REF].

Anthropogenic particulate Fe emissions are of additional concern because of their higher solubility relative to Fe present in naturally occurring dust. Thus anthropogenic particulate Fe is more bioavailable than natural Fe dust, and the former can modulate primary productivity in high-nutrient, low chlorophyll (HNLC) ocean regions [START_REF] Ito | Delivery of anthropogenic bioavailable iron from mineral dust and combustion aerosols to the ocean[END_REF]. Anthropogenic sources are responsible for approximately 7% of the 8.4 Mt of Fe deposited to oceans annually, and therefore have significant impacts upon carbon dioxide fixation globally [START_REF] Wang | Sources, transport and deposition of iron in the global atmosphere[END_REF].

The density and diversity of sources and emission profiles in Dunkirk complicates source apportionment efforts based on off-line analysis significantly [START_REF] Alleman | PM10 metal concentrations and source identification using positive matrix factorization and wind sectoring in a French industrial zone[END_REF]. One technique that is well suited to providing information on the size-resolved chemical composition and mixing state of particles emitted by industrial activities is single particle mass spectrometry.

A review of industrial studies utilising this technique can be found in [START_REF] Riffault | Fine and Ultrafine Particles in the Vicinity of Industrial Activities: A Review[END_REF]. The capability to provide mixing state information for individual particles is key to the role single particle mass spectrometry plays in source apportionment. Ambient particles containing specific internal mixtures of metals and other primary organic and inorganic species have been successfully assigned to their respective sources using this technique. For example, internally mixed Pb/Zn/Cl particles have been assigned to industrial waste incineration (Moffet et al., 2008b), Ga-and Se-containing particles particles have been associated with coal-fired power generation [START_REF] Pekney | Identification of sources of atmospheric PM at the Pittsburgh Supersite, Part I: Single particle analysis and filterbased positive matrix factorization[END_REF][START_REF] Snyder | Estimating the contribution of point sources to atmospheric metals using single-particle mass spectrometry[END_REF] and internally mixed V/Ni/sulfate particles have been associated with ship exhaust [START_REF] Ault | Impact of Emissions from the Los Angeles Port Region on San Diego Air Quality during Regional Transport Events[END_REF][START_REF] Healy | Characterisation of single particles from in-port ship emissions[END_REF]. Fe, Zn and Pbcontaining particles have also been previously assigned to steelwork facility operations [START_REF] J O U R N A L P R E -P R O O F Dall'osto | A Study of the Size Distributions and the Chemical Characterization of Airborne Particles in the Vicinity of a Large Integrated Steelworks[END_REF].

In this context, the NANO-INDUS field campaign was performed in the multi-industry influenced zone of Grande-Synthe in Dunkirk, France, close to a ferromanganese alloy facility and a steelworks [START_REF] Setyan | Investigation on the near-field evolution of industrial plumes from metalworking activities[END_REF]). An aerosol time-of-flight mass spectrometer (ATOFMS)

was deployed to provide size-resolved chemical composition and mixing state data for industrial PM 2.5 with high temporal resolution. The ATOFMS data were used in combination with support J o u r n a l P r e -p r o o f instrumentation measurements to apportion PM 2.5 to local and regional sources and to assess the impact of local metalworking facilities on air quality at the receptor site. Comparison of the ambient ATOFMS data with previous analysis of raw materials (ores) and particles collected on industrial filters from the ferromanganese alloy facility's chimneys [START_REF] Arndt | Scanning electron microscopy-energy dispersive X-ray spectrometry (SEM-EDX) and aerosol time-of-flight mass spectrometry (ATOFMS) single particle analysis of metallurgy plant emissions[END_REF], enabled more effective source assignments for the particle classes observed during the field campaign.

Experimental

Sampling site and Instrumentation

All measurements were made in the industrial area of Grande-Synthe, Dunkirk (51.028809 N, 2.269448 E, 10 m above sea level), approximately 800 m southwest of a ferromanganese alloy manufacturing facility and 1 km southwest of a major steelworks (Figure S1). The adjacency of these two sources precludes separation of their air pollutant contributions based on wind direction alone [START_REF] Setyan | Investigation on the near-field evolution of industrial plumes from metalworking activities[END_REF]. Other nearby sources include a petrochemical storage facility <500 m to the west of the site and urban centres located to the east of the site (Fort-Mardyck and Dunkirk town centre, ~2 km and ~ 7 km from the site, respectively). The field campaign was undertaken from 15 May 2012 to 12 June 2012. An overview of the campaign, local meteorology, the instrumentation deployed and broad PM composition is described in detail elsewhere [START_REF] Setyan | Investigation on the near-field evolution of industrial plumes from metalworking activities[END_REF]. Date and time are reported in Coordinated Universal Time (UTC). PM 2.5 mass concentrations were determined every 2 hours with a MP101M Beta Gauge monitor. 24 to 48-hour filter samples were also collected on 25mm polycarbonate membrane filters (Whatman TM ) using 13-and 3-stage cascade impactors, and weighed with a microbalance (UMT2, Mettler Toledo, Switzerland) [START_REF] Mbengue | Size-distributed metallic elements in submicronic and ultrafine atmospheric particles from urban and industrial areas in northern France[END_REF][START_REF] Mbengue | Metal-bearing fine particle sources in a coastal industrialized environment[END_REF].

The ATOFMS (TSI model 3800) was operated continuously from 18th May -8th June 2012. A detailed description of the ATOFMS can be found elsewhere [START_REF] Gard | Realtime analysis of individual atmospheric aerosol particles: Design and performance of a portable ATOFMS[END_REF]. Briefly, it consists of (i) an aerodynamic focusing lens (TSI AFL100) [START_REF] Su | Development and Characterization of an Aerosol Time-of-Flight Mass Spectrometer with Increased Detection Efficiency[END_REF] used to optimize particle transmission in the aerodynamic diameter (d a ) range 100-3000 nm, (ii) a particle sizing region, and (iii) a bipolar reflectron time-of-flight mass spectrometer. Single particles are desorbed/ionised using a pulsed Nd:YAG laser (λ = 266 nm, ~1 mJ pulse -1 ) and positive and negative ion mass spectra of individual aerosol particles are simultaneously collected for successfully ionised particles. ATOFMS particle counts are scaled to account for sizing and other instrument biases as discussed below.

J o u r n a l P r e -p r o o f 6

ATOFMS Data Analysis

Mass spectral data for over 800,000 single particles was collected using the ATOFMS during the deployment period and was clustered using the K-means algorithm (K=80), as described elsewhere [START_REF] Gross | Environmental chemistry through intelligent atmospheric data analysis[END_REF][START_REF] Healy | Source apportionment of PM2.5 in Cork Harbour, Ireland using a combination of single particle mass spectrometry and quantitative semi-continuous measurements[END_REF][START_REF] Healy | Quantitative determination of carbonaceous particle mixing state in Paris using single-particle mass spectrometer and aerosol mass spectrometer measurements[END_REF]. Clusters with similar mass sepctral profiles, size distributions and temporality were combined. The final merged clusters were then identified as particle "classes"; 33 in total. Approximately 15% of the particles ionised did not produce positive ion mass spectra. The absence of positive ions is likely a result of desorption/ionization matrix effects and precludes the assigment of these particles to a specific category due to the absence of useful marker ions.

The particle class nomenclature used here is regularly used in the literature (Dall'Osto and [START_REF] Dall'osto | Chemical characterisation of single airborne particles in Athens (Greece) by ATOFMS[END_REF]Spencer et al., 2006;[START_REF] Ault | Characterization of the Single Particle Mixing State of Individual Ship Plume Events Measured at the Port of Los Angeles[END_REF][START_REF] Pratt | Mass spectrometry of atmospheric aerosols-Recent developments and applications. Part II: On-line mass spectrometry techniques[END_REF] and typically denotes the dominant species in the positive ion mass spectra. For example, a particle class with high intensities for sodium and elemental carbon is labelled Na-EC. In some cases this is followed by secondary species detected in the negative ion mass spectra (e.g. EC-NH 4 -NO 3 ), referring to elemental carbon internally mixed with ammonium and nitrate.

The ATOFMS does not measure particle number or mass concentrations quantitatively, but instead provides particle counts classified by aerodynamic diameter. The size-dependent transmission efficiency of the focusing lens, data acquisition "busy time", and limited detection of particles <150 nm complicate accurate counting of particles over the ATOFMS size range (100-3000 nm d a ). The desorption/ionization laser used by the ATOFMS also complicates quantitative speciation efforts. Differences in laser output power and variations in power density across the laser beam between pulses [START_REF] Steele | Desorption/Ionization Fluence Thresholds and Improved Mass Spectral Consistency Measured Using a Flattop Laser Profile in the Bioaerosol Mass Spectrometry of Single Bacillus Endospores[END_REF][START_REF] Wenzel | Improvements in ion signal reproducibility obtained using a homogeneous laser beam for on-line laser desorption/ionization of single particles[END_REF] alter the mass of each particle that is effectively desorbed impacting mass spectral signal intensities (Gross et al., 2000a;[START_REF] Reinard | Ion Formation Mechanism in Laser Desorption Ionization of Individual Nanoparticles[END_REF].

Thus the measured ATOFMS counts were scaled using two quantitative particle counting instruments operated concurrently; an optical particle counter (OPC, TSI model 3300) and a scanning mobility particle sizer (SMPS, TSI DMA model 3080 and CPC model 3010). Scaling the ATOFMS number concentration data requires conversion of d a (aerodynamic diameter) into the corresponding volume equivalent diameter (d ve ), using the following relationship:

J o u r n a l P r e -p r o o f

݀ = ఘ ఘ బ ௗ ೡ ఞ (1)
where ρ p is the particle density, assumed for the purposes of number-size distribution scaling to be 1.7 g cm -3 as per previous applications of the method [START_REF] Morawska | Relation between particle mass and number for submicrometer airborne particles[END_REF][START_REF] Reinard | Source characterization and identification by real-time single particle mass spectrometry[END_REF], d ve is the volume equivalent diameter (operationally equivalent to electrical mobility diameter d m measured by the SMPS and optical diameter d o measured by the OPC), ρ 0 is standard density (1 g cm -3 ) and χ represents the dynamic shape factor (e.g. 1, representing spherical shape). This is a simplified version of particle property relationships that are covered in detail elsewhere [START_REF] Decarlo | Particle morphology and density characterization by combined mobility and aerodynamic diameter measurements[END_REF]. Mass concentrations can then be calculated from the scaled number concentrations as follows [START_REF] Reinard | Source characterization and identification by real-time single particle mass spectrometry[END_REF]:

݉ = గ ߩ ݀ ௩ ଷ (2)
The accuracy of the conversion of number concentrations to mass concentrations is dependent on the representativeness of χ and ρ p for each type of particle. In the absence of morphology data χ is assumed to be 1. The use of a single density value, ρ p , for conversion of ATOFMS scaled particle number concentrations to mass concentrations has previously resulted in good agreement with other quantitative measurements [START_REF] Qin | Comparison of Two Methods for Obtaining Quantitative Mass Concentrations from Aerosol Time-of-Flight Mass Spectrometry Measurements[END_REF][START_REF] Healy | Quantitative determination of carbonaceous particle mixing state in Paris using single-particle mass spectrometer and aerosol mass spectrometer measurements[END_REF][START_REF] Zhou | A field measurement based scaling approach for quantification of major ions, organic carbon, and elemental carbon using a single particle aerosol mass spectrometer[END_REF].

However, this simplification is not entirely accurate because different particle types exhibit different densities depending upon their chemical composition (Spencer and Prather, 2006;[START_REF] Reinard | Source characterization and identification by real-time single particle mass spectrometry[END_REF]. A range of more representative densities, estimated from the bulk densities of the chemical components observed in the mass spectra [START_REF] Bein | Identification of sources of atmospheric PM at the Pittsburgh Supersite--Part II: Quantitative comparisons of single particle, particle number, and particle mass measurements[END_REF][START_REF] Reinard | Source characterization and identification by real-time single particle mass spectrometry[END_REF], was therefore used to convert the scaled number-size distributions to mass concentrations for each particle class, as listed in Table S1. ATOFMS-derived mass concentrations for PM 2.5

were averaged to hourly and 24 to 48-hour time resolutions for comparison with PM 2.5 measured concurrently on-line using the beta gauge instrument and off-line through gravimetric filter analysis, respectively. Similar ATOFMS-based source apportionment techniques have been performed previously to separate local and regional contributions to ambient PM in Paris, France and the apportionment results were found to agree well with those modelled using a regional chemical transport model [START_REF] Healy | Quantitative determination of carbonaceous particle mixing state in Paris using single-particle mass spectrometer and aerosol mass spectrometer measurements[END_REF][START_REF] Skyllakou | Contributions of local and regional sources to fine PM in the megacity of Paris[END_REF].

Results and Discussion

ATOFMS Results Overview

J o u r n a l P r e -p r o o f ATOFMS-derived mass concentrations of PM 2.5 were in reasonable temporal agreement with concurrently measured on-line beta gauge PM 2.5 mass concentrations at 2-hour resolution (R 2 = 0.62) and in good temporal agreement with gravimetric PM 2.5 mass concentrations at 24 to 48hour resolution (R 2 = 0.91) as shown in Figure 1. This temporal agreement is in line with previous studies involving similar scaling procedures for single particle mass spectrometers (R 2 = 0.64-0.91) [START_REF] Qin | Comparison of Two Methods for Obtaining Quantitative Mass Concentrations from Aerosol Time-of-Flight Mass Spectrometry Measurements[END_REF][START_REF] Jeong | Quantification of aerosol chemical composition using continuous single particle measurements[END_REF][START_REF] Healy | Quantitative determination of carbonaceous particle mixing state in Paris using single-particle mass spectrometer and aerosol mass spectrometer measurements[END_REF][START_REF] Zhou | A field measurement based scaling approach for quantification of major ions, organic carbon, and elemental carbon using a single particle aerosol mass spectrometer[END_REF] and indicates that changes in ATOFMS-derived mass concentrations are representative of variations in ambient PM 2.5 mass concentrations at the site during the field study.

The 33 distinct ATOFMS particle classes identified through clustering were grouped into ten general "categories" for simplicity based on the dominant elements in their respective mass spectra. Briefly, the categories identified are: K-rich, EC-rich (elemental carbon), Na-rich, Ferich, Ca-rich, V-rich, Mn-rich, OC-rich (organic carbon), Pb-rich/Li-rich and Others. The relative contributions of each category in terms of mass concentration is shown in Table S1. The dominant categories are K-rich (24%), EC-rich (23%) Na-rich (10%) and Fe-rich (10%), followed by OC-rich (4%), Ca-rich (4%), V-rich (4%), Mn-rich (3%) and Pb-rich/Li-rich (3%).

The "Others" category contains particle classes with minimal mass contributions (Sulfur and Amines, <1%). Temporal trends for each category are shown in Figure 2. It is clear from these temporal trends that the Na-rich, K-rich and EC categories exhibit prolonged, persistent mass contributions, consistent with regional background sources. Briefly, the K-rich category is dominated by a single particle class (K-NO 3 ) often associated with regional biomass combustion [START_REF] Silva | Size and chemical characterization of individual particles resulting from biomass burning of local Southern California species[END_REF], routinely observed in regional background air in ATOFMS datasets collected across Europe [START_REF] Dall'osto | Real time chemical characterization of local and regional nitrate aerosols[END_REF][START_REF] Healy | Single-particle speciation of alkylamines in ambient aerosol at five European sites[END_REF][START_REF] Arndt | Sources and mixing state of summertime background aerosol in the north-western Mediterranean basin[END_REF]. The ATOFMS is highly sensitive to potassium (Gross et al., 2000a), and while the potassium ion is a dominant feature of biomass burning particle mass spectra, it is typically present as a trace component, with EC, OC and secondary inorganics dominating the composition in terms of mass [START_REF] Healy | Quantitative determination of carbonaceous particle mixing state in Paris using single-particle mass spectrometer and aerosol mass spectrometer measurements[END_REF]. Particles in the EC-rich category are associated with various combustion sources, including vehicle exhaust and biomass burning (Spencer et al., 2006;Moffet et al., 2008a), and are often observed to be internally mixed with OC and secondary inorganics [START_REF] Cahill | The mixing state of carbonaceous aerosol particles in northern and southern California measured during CARES and CalNex 2010[END_REF][START_REF] Healy | Sources and mixing state of size-resolved elemental carbon particles in a European megacity: Paris[END_REF]. The Na-rich category is predominantly comprised of supermicron sea salt internally mixed with nitrate, which is expected for a coastal urban site [START_REF] Gard | Direct observation of heterogeneous chemistry in the atmosphere[END_REF]. These three major particle categories are routinely observed in ATOFMS field studies. However,

Metalworking Particle Source Identification

Temporal trends for individual particle classes exhibiting elevated concentrations during the prolonged periods when the site was downwind of the ferromanganese facility and the steelworks [START_REF] Setyan | Investigation on the near-field evolution of industrial plumes from metalworking activities[END_REF] are shown in Figure 2. These include metal-rich particles with a variety of chemical mixing states and particles containing OC and polycyclic aromatic hydrocarbons (PAHs).

Mn-rich particles:

Three distinct Mn-containing particle classes were identified. Mn-K particles were characterized by the presence of Mn + (m/z 55) and K + (m/z 39) while Mn-Al-K particles exhibited a strong additional signal for aluminium (Al + , m/z 27) and Mn-K-Fe particles exhibited a strong additional signal for iron (Fe + , m/z 56) Average mass spectra for the three Mn-containing classes are shown in Figure 3, and the positive ion mass spectra are consistent with ATOFMS positive J o u r n a l P r e -p r o o f ion mass spectra for particles collected at the ferromanganese facility [START_REF] Arndt | Scanning electron microscopy-energy dispersive X-ray spectrometry (SEM-EDX) and aerosol time-of-flight mass spectrometry (ATOFMS) single particle analysis of metallurgy plant emissions[END_REF] (Figure S2). Mn-K particles are consistent with samples of the ores used at the facility as well as filter samples from the cooling chimney. Mn-Al-K particles also resemble both ore samples and the filter samples from the chimneys. Finally, Mn-K-Fe particle mass spectra are consistent with mass spectra for ore samples exclusively. Ambient Mn-K, Mn-Al-K and Mn-K-Fe are all enriched in nitrate (NO 2 -/NO 3 -, m/z -46/-62) and chloride (Cl -, m/z -35) relative to the facility samples [START_REF] Arndt | Scanning electron microscopy-energy dispersive X-ray spectrometry (SEM-EDX) and aerosol time-of-flight mass spectrometry (ATOFMS) single particle analysis of metallurgy plant emissions[END_REF], likely due to secondary accumulation of inorganic ions either through mixing with either inorganic gases from the metalworking facilities themselves or with ambient air after emission. All three Mn-rich particle classes are therefore assigned to the ferromanganese facility, which has been previously identified as the major local source of Mn in a previous apportionment study based on off-line filter analysis [START_REF] Alleman | PM10 metal concentrations and source identification using positive matrix factorization and wind sectoring in a French industrial zone[END_REF].

Fe-rich particles:

Mass spectra for particles containing internally mixed iron, potassium and calcium (Fe-K-Ca) feature a dominant signal for Fe + (m/z 56) followed by K + (m/z 39), Ca + (m/z 40), Na + (m/z 23)

and Al + (m/z 27) (Figure 4). Negative ion mass spectra contain peaks corresponding to chloride [START_REF] Flament | Iron isotopic fractionation in industrial emissions and urban aerosols[END_REF]. Iron-rich particles similar to the ATOFMS Fe-K-Ca and Fe-K-Na classes were also detected by ATOFMS near a steelworks in Port Talbot (UK) [START_REF] J O U R N A L P R E -P R O O F Dall'osto | A Study of the Size Distributions and the Chemical Characterization of Airborne Particles in the Vicinity of a Large Integrated Steelworks[END_REF], where they accounted for 3% of the particles detected and were attributed to the hot and cold mills. In the absence of any unique ions to differentiate between iron-containing particles from the steelworks and those emitted by the ferromanganese facility, and given that Fe/K/Ca particles only contributed a small fraction of the particles identified in the ferromanganese facility samples [START_REF] Arndt | Scanning electron microscopy-energy dispersive X-ray spectrometry (SEM-EDX) and aerosol time-of-flight mass spectrometry (ATOFMS) single particle analysis of metallurgy plant emissions[END_REF], it is likely that the J o u r n a l P r e -p r o o f majority of iron-containing particles detected during the field study originated from the steelworks. [START_REF] Setyan | Investigation on the near-field evolution of industrial plumes from metalworking activities[END_REF] found that Mn/Fe elemental ratios in filter samples collected during the periods when the receptor site was downwind of the metalworking facilities were between the ratios expected for emissions from the ferromanganese facility and the steelworks, suggesting that emissions from both facilities were impacting the site. Contributions of Fe from both facilities were also observed in the previous source apportionment study [START_REF] Alleman | PM10 metal concentrations and source identification using positive matrix factorization and wind sectoring in a French industrial zone[END_REF].

K-rich particles:

As shown in Figure 5, K-Cl particles are characterized by peaks for K + (m/z 39), Na + (m/z 23),

Cl -(m/z -35), and nitrate (NO 2 -/NO 3 -, m/z -46/-62). The local steelworks emits potassium chloride during the sintering process [START_REF] Setyan | Investigation on the near-field evolution of industrial plumes from metalworking activities[END_REF], and potassium chloride emissions are estimated to represent ~40% of total sintering emissions from the facility [START_REF] Flament | Iron isotopic fractionation in industrial emissions and urban aerosols[END_REF][START_REF] Hleis | Chemical profile identification of fugitive and confined particle emissions from an integrated iron and steelmaking plant[END_REF]. K-Cl particles were not observed in the chimney filter or ore samples from the ferromanganese facility [START_REF] Arndt | Scanning electron microscopy-energy dispersive X-ray spectrometry (SEM-EDX) and aerosol time-of-flight mass spectrometry (ATOFMS) single particle analysis of metallurgy plant emissions[END_REF]. K-Cl particles have however, been detected downwind of the Port Talbot steelworks [START_REF] J O U R N A L P R E -P R O O F Dall'osto | A Study of the Size Distributions and the Chemical Characterization of Airborne Particles in the Vicinity of a Large Integrated Steelworks[END_REF] (Figure 5). No aluminium signal was observed in K-Al-Si positive mass spectra, likely due to matrix-related ion suppression [START_REF] Reinard | Ion Formation Mechanism in Laser Desorption Ionization of Individual Nanoparticles[END_REF]. K-Cl and K-Al-Si particles were not observed in the ferromanganese facility samples and are therefore assigned to the steelworks.

However, K-Al particles were positively identified in the ferromanganese facility firing chimney samples (Figure S4). ), consistent with the firing chimney samples, however this ion is absent in the ambient Ca-K particle spectra, indicating that ambient Ca-K particles may be associated with the steelworks instead. [START_REF] Flament | Iron isotopic fractionation in industrial emissions and urban aerosols[END_REF] noted that calcium salts, including calcite and lime, are used in the sintering process at the steelworks, however the Ca-containing particles produced in that case are aggregates of Fe 2 O 3 , CaCO 3 and KCl. Cacontaining particles constituted only 8% of the particles analyzed from the steelworks sintering process by [START_REF] Flament | Iron isotopic fractionation in industrial emissions and urban aerosols[END_REF], while Ca-rich particles accounted for approximately 80% of the particles in the firing chimney filter sample [START_REF] Arndt | Scanning electron microscopy-energy dispersive X-ray spectrometry (SEM-EDX) and aerosol time-of-flight mass spectrometry (ATOFMS) single particle analysis of metallurgy plant emissions[END_REF]. Ca-containing particles have also been assigned to the blast furnace unit of the Port Talbot steelworks [START_REF] Taiwo | A review of receptor modelling of industrially emitted particulate matter[END_REF], but those particles were also internally mixed with OC, and no Ca particles were reported in the previous Port Talbot steelworks studies by [START_REF] J O U R N A L P R E -P R O O F Dall'osto | A Study of the Size Distributions and the Chemical Characterization of Airborne Particles in the Vicinity of a Large Integrated Steelworks[END_REF],2012). Thus, while there remains a possibility that some of the Ca-rich particles could have originated from the steelworks, the ferromanganese facility is likely the dominant source.

OC-rich particles:

Three OC-containing particle classes were identified, of which the OC class was the dominant contributor in terms of mass. Average mass spectra are shown in Figure 7. All three classes are characterized by the presence of hydrocarbon fragments in the positive mass spectra mixed sulfate is consistent with emissions from the coke plant at the steelworks [START_REF] Zhang | Near real-time PM1 chemical composition measurements at a French urban background and coastal site under industrial influence over more than a year: Temporal variability and assessment of sulfur-containing emissions[END_REF]. Mass spectra similar to the three OC-containing classes reported here were also observed J o u r n a l P r e -p r o o f downwind of the Port Talbot steelworks [START_REF] Dall'osto | Real-Time Measurements of Nonmetallic Fine Particulate Matter Adjacent to a Major Integrated Steelworks[END_REF], with m/z 202 assigned to pyrene/fluoranthene as per previous ATOFMS field studies (Gross et al., 2000b). In the case of Dall'Osto et al. ( 2012) OC-PAH particles were attributed to the hot and cold mills, while OC and PAH particles were related to the steel/coke-making sector. Thus, all three organic carboncontaining classes are attributed to the steelworks here.

(C 2 H 3+ /C 3 H + /C 2 H 3 O + /C 4 H 3 + /C 5 H 3 +/C 6 H 2 + /C 7 H 3 + , m/z

Pb-rich and Li-rich particles:

Average ATOFMS mass spectra for Pb-rich particles are shown in [START_REF] J O U R N A L P R E -P R O O F Guazzotti | Characterization of carbonaceous aerosols outflow from India and Arabia: Biomass/biofuel burning and fossil fuel combustion[END_REF][START_REF] Furutani | Singleparticle chemical characterization and source apportionment of iron-containing atmospheric aerosols in Asian outflow[END_REF]. Zn-containing particles were also associated with the Port Talbot steelworks, but those particles did not contain internally mixed Li [START_REF] J O U R N A L P R E -P R O O F Dall'osto | A Study of the Size Distributions and the Chemical Characterization of Airborne Particles in the Vicinity of a Large Integrated Steelworks[END_REF]. Li-Zn particles have a minimal mass contribution overall and it cannot be easily determined whether they originate from the ferromanganese facility or the steelworks.

Source Apportionment

Summing the mass concentrations of the different particle classes associated with the ferromanganese facility and the steelworks enabled an estimation of the PM 2.5 associated with metalworking activities at the receptor site at hourly resolution. Elevated metalworking PM 2.5 concentrations are strongly dependent on northeasterly wind directions, consistent with the locations of the two metalworking facilities (including ore and coal stockpiles), as shown in J o u r n a l P r e -p r o o f 14 Figure 9. Elevated non-metalworking (background) PM 2.5 mass concentrations are observed under northerly, easterly and southerly wind directions, consistent with more diffuse regional background sources. The contribution of metalworking activities to PM 2.5 at the receptor site ranged from 1-65% over the entire measurement period, with an average contribution of 17%.

The average contribution for those periods when wind was predominantly from the northeast (shaded grey periods in Figure 2) is higher, at 28%. The overall average contribution of 17% is somewhat lower than the 28% contribution these two facilities were estimated to contribute to PM 10 at a site approximately 3 km to the southeast through PMF analysis of filters collected in 2003[START_REF] Steele | Desorption/Ionization Fluence Thresholds and Improved Mass Spectral Consistency Measured Using a Flattop Laser Profile in the Bioaerosol Mass Spectrometry of Single Bacillus Endospores[END_REF][START_REF] Alleman | PM10 metal concentrations and source identification using positive matrix factorization and wind sectoring in a French industrial zone[END_REF]. Although the local metalworking facilities are not the dominant contributor to PM 2.5 mass at the receptor site, their dominant contributions to toxic metal and PAH-containing particle concentrations is of concern from an exposure perspective [START_REF] Schroeder | Toxic Trace Elements Associated with Airborne Particulate Matter: A Review[END_REF][START_REF] Durant | Human cell mutagenicity of oxygenated, nitrated and unsubstituted polycyclic aromatic hydrocarbons associated with urban aerosols[END_REF][START_REF] Majestic | Development of a Manganese Speciation Method for Atmospheric Aerosols in Biologically and Environmentally Relevant Fluids[END_REF][START_REF] Zhang | Source Apportionment of in Vitro Reactive Oxygen Species Bioassay Activity from Atmospheric Particulate Matter[END_REF][START_REF] Hu | Bioaccessibility and health risk of arsenic and heavy metals (Cd, Co, Cr, Cu, Ni, Pb, Zn and Mn) in TSP and PM2.5 in Nanjing, China[END_REF]. The metalworking facilities were also the dominant contributor to ultrafine particles at the site [START_REF] Setyan | Investigation on the near-field evolution of industrial plumes from metalworking activities[END_REF], a size fraction receiving increased attention with respect to its potential toxicological impacts [START_REF] Ohlwein | Health effects of ultrafine particles: a systematic literature review update of epidemiological evidence[END_REF][START_REF] Saleh | Exposure to Atmospheric Ultrafine Particles Induces Severe Lung Inflammatory Response and Tissue Remodeling in Mice[END_REF][START_REF] Sotty | Toxicological effects of ambient fine (PM2.5-0.18) and ultrafine (PM0.18) particles in healthy and diseased 3D organo-typic mucocilary-phenotype models[END_REF].

Conclusions

The unique particle mixing states associated with local metalworking facilities identified at the Dunkirk receptor site will be useful for future researchers aiming to apportion particulate matter to metalworking sources in other locations globally. The facilities investigated here use industrial processes that are common in the pyrometallurgic industry (sintering, smelting/reduction in an electric/blast furnace, refining, rolling and crushing of finished product) and therefore can be considered representative of that sector. The number of distinct chemical mixing states identified highlights the complexity and diversity in primary particle composition associated with emissions from metalworking. Particles containing Mn and Ca were associated predominantly with the ferromanganese facility, while particles containing Fe, K, Pb and PAHs were associated predominantly with the steelworks. Internally mixed fluoride was also found to be a useful discriminating marker for identifying ferromanganese firing chimney emissions specifically. The two metalworking facilities combined are estimated to contribute 17% on average to PM 2.5 mass concentrations at the receptor site during the measurement period, but are the dominant contributors of particles containing transition metals and PAHs; species associated with negative

) Ca Ca-K OC OC-PAH PAH K-Cl K-Al-Si K-Al Fe-K-Na Fe-K-Ca Mn-K Mn-Al-K Mn-K-Fe Pb-K Li-Zn
J o u r n a l P r e -p r o o f speed < 1 m s -1 are not included in the analysis. Plots generated using the Zefir Toolkit [START_REF] Petit | A user-friendly tool for comprehensive evaluation of the geographical origins of atmospheric pollution: Wind and trajectory analyses[END_REF] running in Igor Pro v6.37 (Wavemetrics Inc.).

b

  now at: ETH Zürich, Institute of Environmental Engineering, 8093 Zürich, Switzerland c now at: CzechGlobe -Global Change Research Institute, Czech Academy of Sciences, 60300, Brno, Czech Republic 1 Introduction

J

  o u r n a l P r e -p r o o f several of the less abundant particle classes were found to be associated with local metalworking activities.As outlined by[START_REF] Setyan | Investigation on the near-field evolution of industrial plumes from metalworking activities[END_REF], two distinct meteorological sea-breeze periods when winds were predominantly from the northeast were identified during the NANO-INDUS field campaign when the sampling site was downwind of the nearby ferromanganese and steelworks facilities (May 25-28 and June 02-03). During these events, particle number concentrations increased considerably, with concurrent increases in non-refractory chloride, organic aerosol and metals including manganese and iron. As shown in Figure2, concentrations of several metal-rich particle categories detected by the ATOFMS were enhanced during these periods (shaded in grey). Particle classes associated with the ferromanganese alloy facility were identified through comparison with previous scanning electron microscopy/electron dispersive X-ray spectrometry (SEM-EDX) and ATOFMS analysis of raw materials (ores) and cooling, firing and smelting chimney filter samples previously collected at the facility[START_REF] Arndt | Scanning electron microscopy-energy dispersive X-ray spectrometry (SEM-EDX) and aerosol time-of-flight mass spectrometry (ATOFMS) single particle analysis of metallurgy plant emissions[END_REF]. Particle classes were assigned to the steelworks based on previous off-line elemental analysis of particles collected downwind of this facility[START_REF] Mbengue | Metal-bearing fine particle sources in a coastal industrialized environment[END_REF] and previous ATOFMS measurements made in the vicinity of a large steelworks in the UK[START_REF] J O U R N A L P R E -P R O O F Dall'osto | A Study of the Size Distributions and the Chemical Characterization of Airborne Particles in the Vicinity of a Large Integrated Steelworks[END_REF][START_REF] Dall'osto | Real-Time Measurements of Nonmetallic Fine Particulate Matter Adjacent to a Major Integrated Steelworks[END_REF].

(-

  z -46/-62). Fe-K-Na particles are quite similar to Fe-K-Ca particles, differing only in the intensities of sodium and potassium relative to iron. The mass spectral features of both particle classes are similar to internally mixed Fe/K/Ca particles collected from the chimneys of the ferromanganese facility[START_REF] Arndt | Scanning electron microscopy-energy dispersive X-ray spectrometry (SEM-EDX) and aerosol time-of-flight mass spectrometry (ATOFMS) single particle analysis of metallurgy plant emissions[END_REF]) (FigureS3), although the ambient particles shown in Figure4exhibit higher signals for nitrate (NO 2 -46/-62), again indicating uptake of nitrate after emission. While similarities exist between ambient and ferromanganese facility sample mass spectra for Fe-containing particles, the nearby steelworks remains the largest source of particulate iron in Dunkirk. A previous analysis of the steelworks particulate emissions demonstrated that most particles emitted were comprised of either Fe 2 O 3 , CaCO 3 , KCl or were graphite flakes

  and are thus assigned to the steelworks here. K-Al-Si particles are characterized by signals for K + (m/z 39), Na + (m/z 23), Fe + (m/z 56), AlO -(m/z -43), AlO(OH) -z -79), and chloride (Cl -, m/z -35/-37)

--

  Ambient K-Al particles produced stronger signals for nitrate (NO 2 z -97) but both the ambient and the firing chimney K-Al particles exhibited similar signals for F -(m/z -19). Therefore fluoride represents a useful discriminatory marker for identifying potassium-containing particles from the ferromanganese facility. Ca-rich particles: J o u r n a l P r e -p r o o f Ca particles are characterized by a strong signal for Ca + (m/z 40) and smaller signals for Na + (m/z 23) and CaO + (m/z 56), while Ca-K particles contain stronger signals for Na + (m/z 23) and K +(m/z 39) as shown in Figure6. Based on comparison with the ferromanganese facility samples, both Ca and Ca-K particles appear to originate from the firing chimney, although ambient Ca-K particles exhibit higher signals for Na than the chimney samples (FigureS3). Negative ion mass spectra for both ambient Ca and Ca-K particles also exhibited higher signals for nitrate (NO 2 z -97) than their respective chimney sample particles again likely due to secondary accumulation after emission. A signal for fluoride is present in the ambient Ca negative ion mass spectra (F -, m/z -19

  27/37/43/51/63/74/87) together with K + (m/z 39) and Na + (m/z 23). Carbon fragment ions (C -z -46/-62) is dominant in the OC and PAH classes. The presence of internally

--

  Figure 8, characterized by internally mixed K + (m/z 39), Pb + (m/z 208), Cl -(m/z -35) and nitrate (NO 2 z -46/-62).No equivalent particles were observed in the ferromanganese facility samples. Pb-containing particles similar to the Pb-rich class were also observed in the Port Talbot study, and represented <1% of the particles ionized. In that case Pb-containing particles were assigned to the hot and cold mills[START_REF] J O U R N A L P R E -P R O O F Dall'osto | A Study of the Size Distributions and the Chemical Characterization of Airborne Particles in the Vicinity of a Large Integrated Steelworks[END_REF]. The Port Talbot Pb-containing particles were internally mixed with KCl, which is also the case for Dunkirk as shown in Figure8. Thus Pb-rich particles are attributed to the steelworks here.Li-Zn particles contain Li + (m/z 7), Na + (m/z 23), K + (m/z 39), Zn + (m/z 65/66/68), nitrate (NO 2 z -79), as shown in Figure8, and were not observed in the ferromanganese facility samples. Particles detected by ATOFMS containing Li have been assigned to coal combustion sources in previous studies (

JFigures 718 Figure 1 :Figure 2 :

 71812 Figures 718

Figure 3 :

 3 Figure 3: Average mass spectra for Mn-rich metalworking particle classes. 736

Figure 4 :

 4 Figure 4: Average mass spectra for iron-rich metalworking particle classes.

  r n a l P r e -p r o o f

Figure 5 :Figure 6 :

 56 Figure 5: Average mass spectra for K-rich metalworking particle classes. 744

Figure 8 :

 8 Figure 7: Average mass spectra for OC-rich metalworking particle classes. 753

Figure 9 :

 9 Figure 9: Dependence of concentrations of metalworking and background ATOFMS-derived PM 2.5 mass concentrations upon local wind speed (m s -1 ) and wind direction. Periods with wind

  r n a l P r e -p r o o f Highlights Particles assigned to a ferromanganese facility and a steelworks based on composition Metalworking particle composition and mixing state is complex and diverse Metalworking emissions contributed 17% to PM 2.5 on average at the receptor site Metalworking is the dominant source of metals and polycyclic aromatic hydrocarbons J o u r n a l P r e -p r o o f

J o u r n a l P r e -p r o o f

Acknowledgements

Funding for the field measurements was provided by the French Agency of Environment and Energy Management (ADEME Grant #1181C0089). LPCA and IMT Lille Douai acknowledge financial support from the CaPPA (Chemical and Physical Properties of the Atmosphere) project funded by the French National Research Agency (ANR) through the PIA (Programme d'Investissement d'Avenir) under contract ANR-11-LABX-0005-01, and two CPER projects funded by the French Ministry of Higher Education and Research, the CNRS, the Regional Council "Hauts-de-France" and the European Regional Development Fund (ERDF): Climibio, and IRENI (additionally financed by the Communauté Urbaine de Dunkerque). We thank Vale Manganese France (VMF) for their collaboration in this study and Marc Fourmentin for J o u r n a l P r e -p r o o f

health outcomes. The single particle source apportionment approach used here is also expected to be useful in other regions where the relative contributions of industrial emissions, including metalworking emissions, to ambient PM 2.5 concentrations need to be assessed. Knowledge of these contributions is particularly valuable for designing effective local air quality strategies.

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: