Andréas Heinen
email: andreas.heinen@u-cergy.fr

Alfonso Valdesogo

Spearman rank correlation of the bivariate Student t and scale mixtures of normal distributions

Keywords: Rank correlation, Rank-based estimation, Scale mixture of normals, Student t, Spearman's rho. 2010 MSC: Primary 62H20, Secondary 62H12

We derive an expression for the Spearman rank correlation of bivariate scale mixtures of normals (SMN) and we show that within this class, for any value of the correlation parameter, the Spearman rank correlation of the normal is the greatest in absolute value. We then provide expressions for the symmetric generalized hyperbolic, the Bessel, and the Laplace distributions. We further derive an expression for the Spearman rank correlation of the Student t distribution in terms of an easily computable one-dimensional integral, and we also consider the special case of the Cauchy. Finally, we show how our results can be used in a rank-based estimation of the parameters of the Student t distribution.

Introduction

The rank correlations of the bivariate normal distribution are well-known, see [START_REF]Ordinal measures of association[END_REF] and [START_REF] Cramér | Mathematical Methods of Statistics[END_REF], who attributes the result to [START_REF] Sheppard | On the application of the theory of error to cases of normal distribution and normal correlation[END_REF]. The Kendall and Spearman rank correlations of the bivariate normal with correlation r are, respectively,

τ N (r) = 2 π arcsin(r), ρ N (r) = 6 π arcsin r 2 .
As shown by [START_REF] Lindskog | Kendall's tau for elliptical distributions[END_REF], the relation between Kendall's tau and the linear correlation parameter holds more generally for all elliptical distributions with continuous marginals, including the bivariate Student t. However, as shown by [START_REF] Hult | Multivariate extremes, aggregation and dependence in elliptical distributions[END_REF], this invariance does not hold for Spearman's rho, which, to the best of our knowledge is not known in closed form for scale mixtures of normals, an important subclass of the elliptical distributions, even for a very common distribution such as the Student t (for a general discussion of dependence in meta-elliptical distributions, see [START_REF] Abdous | Dependence properties of meta-elliptical distributions[END_REF][START_REF] Fang | The meta-elliptical distributions with given marginals[END_REF]).

In this paper we first express the Spearman rank correlation of bivariate scale mixtures of normals (SMNs, see [START_REF] Andrews | Scale mixtures of normals distributions[END_REF]) as an expectation of a correlation mixture of the Spearman rank correlation of the Gaussian. We show that within this class, for any value of the correlation parameter, the Spearman rank correlation of the normal is the greatest in absolute value. We then provide expressions for the symmetric generalized hyperbolic, the Bessel, and the Laplace distributions. We further derive an expression for the Spearman rank correlation of the Student t distribution in terms of an easily computable one-dimensional integral, and we also consider the special case of the Cauchy. Finally, we show how our results can be used in a rank-based estimation of the parameters of the Student t distribution.

Scale mixtures of normals

Scale mixtures of normals are a subclass of the elliptical distributions. A bivariate scale mixture of normals X ∼ S MN 2 (µ, Σ, f W ) can be expressed as the product of a normal vector Z ∼ N 2 (0, Σ), and the square root of a scalar-valued random variable W with positive support and density f W , stochastically independent of Z:

X = µ + √ W Z,
where N n (0, Σ) is the n-dimensional normal distribution with mean 0 and variance-covariance Σ and µ is a vector location parameters. Conditionally on W, X is Gaussian:

X|W = w ∼ N 2 (µ, wΣ).
The density of X then follows by integration with respect to the density of the mixing variable W:

f X (x 1 , x 2 ) = ∞ 0 f X|W (x 1 , x 2 |w) f W (w)dw,
where f X|W is the density of the bivariate normal. By Sklar's theorem (see [START_REF] Sklar | Fonctions de répartition à n dimensions et leurs marges[END_REF]) the joint cumulative distribution function F X of the bivariate SMN variable X can be expressed in terms of a copula C, which captures the underlying dependence structure, and of its marginals F i , i ∈ {1, 2}:

F X (x 1 , x 2 ) = C{F 1 (x 1 ), F 2 (x 2 )}.
The Kendall and Spearman rank correlations, τ and ρ, depend on the copula but not on the marginal distributions (see, e.g., [START_REF] Nelsen | An Introduction to Copulas[END_REF]):

τ = 4 1 0 1 0 C(u 1 , u 2 )dC(u 1 , u 2 ) -1, ρ = 12 1 0 1 0 C(u 1 , u 2 )du 1 du 2 -3.
Thus we limit attention to the case X ∼ S MN 2 (0, P, f W ) when Z follows a standardized normal with unit variances, correlation matrix P, and correlation r. Whenever W has a finite mean, X has finite second moments, and thus r is the correlation of X. Throughout the paper, we will refer to r as the correlation parameter, even when X does not have finite second moments.

A number of well-known distributions can be written as scale mixtures of normals. The most general distribution we consider is the bivariate symmetric generalized hyperbolic distribution X ∼ GH 2 (λ, χ, ψ, P), which depends on the correlation parameter r through the matrix P, a parameter λ that influences tail behavior, a scaling parameter χ and a shape parameter ψ. It is a special case of the multivariate generalized hyperbolic distribution introduced in [START_REF] Barndorff-Nielsen | Exponentially decreasing distributions for the logarithm of particle size[END_REF][START_REF] Barndorff-Nielsen | Hyperbolic distributions and distributions on hyperbolae[END_REF] and it has been also referred to as the generalized multivariate modified Bessel (see [START_REF] Thabane | Hypothesis testing for the generalized multivariate modified Bessel model[END_REF]). Its density is

f X (x) = |P| -1/2 ( √ χψ) -λ ψ 2πK λ ( √ χψ) K λ-1 ψ(χ + x ⊤ P -1 x) ψ(χ + x ⊤ P -1 x) (1-λ) ,
where K λ (z) is a modified Bessel function of the third kind:

K λ (z) = 1 2 ∞ 0 y λ-1 exp - 1 2 z(y + y -1 ) dy.
It can be written as a scale mixture of normals when W follows a generalized inverse Gaussian, W ∼ GIG(λ, χ, ψ), with density

f W (w) = χ -λ √ χψ λ 2K λ √ χψ w λ-1 exp - 1 2 χw -1 + ψw .
The bivariate generalized hyperbolic distribution includes as special cases the hyperbolic when λ = 3/2 (see [START_REF] Blaesild | The two-dimensional hyperbolic distribution and related distributions, with an application to johannsen's bean data[END_REF]), the normal inverse gamma (NIG) when λ = -1/2, and the bivariate Bessel distribution of [START_REF] Madan | The variance gamma (V.G.) model for share market returns[END_REF] (the univariate Bessel dates back to [START_REF] Mckay | A Bessel function distribution[END_REF]), also known as variance-gamma or generalized Laplace, with parameter q when λ = q + 1, χ = 0, and where, in order to have unit variances, we further impose ψ = 2(q + 1). The density of the bivariate Bessel

X ∼ B 2 (q, P) is f X (x) = 1 π |P| -1/2 q + 1 Γ(1 + q)2 q 2(q + 1)x ⊤ P -1 x q K q 2(q + 1)x ⊤ P -1 x .
It can be written as a scale mixture of normals when W ∼ Γ(q + 1, q + 1), where the density of a Gamma variable

Γ(α, β) is f W (w) = β α Γ(α) x α-1 exp (-βw).
When we further impose q = 0, we obtain the bivariate symmetric Laplace distribution X ∼ L 2 (P), with correlation matrix P, which corresponds to W ∼ Γ(1, 1), or alternatively a mean one exponential W ∼ E (1). Its density is

f X (x) = |P| -1/2 π K 0 2x ⊤ P -1 x .
The generalized hyperbolic distribution also includes the Pearson type VII (see [START_REF] Fernndez | Bayesian regression analysis with scale mixtures of normals[END_REF]) when λ = -ν/2 and ψ = 0, whose density is

f X (x) = 1 2π ν χ |P| -1/2 1 + x ⊤ P -1 x χ -(ν/2+1)
.

It can be written as a scale mixture of normals when W ∼ IG(ν/2, χ/2), and the density of the inverse Gamma

IG(α, β) is f W (w) = β α Γ(α) w -(α+1) exp (-β/w).
When λ = -ν/2, χ = ν and ψ = 0, W ∼ IG(ν/2, ν/2), it reduces to the Student t with degrees of freedom ν, X ∼ T 2 (P, ν), whose density is

f X (x) = 1 2π |P| -1/2 1 + x ⊤ P -1 x ν -(ν/2+1)
.

With unit degrees of freedom ν = 1, W ∼ IG(1/2, 1/2) and we obtain the bivariate Cauchy, X ∼ C 2 (P):

f X (x) = 1 2π |P| -1/2 1 + x ⊤ P -1 x -3/2 .
3. Spearman's rho of scale mixtures of normals (SMN) Proposition 1 (Spearman's rho for scale mixture of normals). The Spearman rank correlation of a bivariate scale

mixture of normals X = X 1 X 2 ∼ S MN(0, P, f W ) is ρ S MN = 6 π E Ṽ arcsin r Ṽ , (1)
where the mixing density is

f Ṽ (ṽ) = ∞ 0 u 1 /ṽ u 1 ṽ 4 (u 1 u 2 ) 4 f W ṽ u 1 u 2 f W       u 1 -u 2 ṽ u 1 u 2 2       f W       u 2 -u 1 ṽ u 2 1 u 2       du 2 du 1 , (2)
when the scaling random variable W has support on [0, ∞). More generally, when the support

of variable W is [a, b] with b > a ≥ 0, integration in (2) is over the domain D = {u 1 , u 2 : a ≤ ṽ/(u 1 u 2 ) ≤ b; a ≤ (u 1 -u 2 ṽ)/(u 1 u 2 2 ) ≤ b; a ≤ (u 2 -u 1 ṽ)/(u 2 1 u 2 ) ≤ b}.
Proof. Let W i , i ∈ {1, 2, 3}, be three i.i.d. variables with density f W . Define a bivariate vector X = X1 X2 , whose components are two independent SMN variables:

Xi |W i = w i ∼ N 1 (0, w i ), i ∈ {1, 2}.
The bivariate SMN vector X can be represented as X|W 3 = w 3 ∼ N 2 (0, w 3 P). Thus X is a bivariate vector of independent SMN variables and X is a bivariate vector of joint SMN variables, independent of X. Combining these two vectors into vector X = X X , we get

X|(W 1 = w 1 , W 2 = w 2 , W 3 = w 3 ) ∼ N 4 (0, Ω),
and

Ω = Ω 11 0 0 w 3 P , Ω 11 = w 1 0 0 w 2 . Define Y = B 0 X, with B 0 = (I 2 , -I 2 )
, where Y is a difference between a bivariate SMN and a bivariate vector of independent SMN variables. Then

Y|(W 1 = w 1 , W 2 = w 2 , W 3 = w 3 ) ∼ N 2 (0, B 0 ΩB 0 ),
where

B 0 ΩB 0 = w 1 + w 3 rw 3 rw 3 w 2 + w 3 ,
and, defining

Ṽ = W 3 {(W 1 + W 3 )(W 2 + W 3 )} -1/2
, the correlation is rṽ, where ṽ is the realization of Ṽ. By definition of Spearman's rho (see, e.g., [START_REF]Ordinal measures of association[END_REF]),

ρ S MN = 12 Pr(X 1 ≤ X1 , X 2 ≤ X2 ) -3 = 12 Pr(Y 1 > 0, Y 2 > 0) -3 = 12E Ṽ E ½ {Y 1 >0,Y 2 >0} | Ṽ -3 = 12 1 4 + 1 2π E Ṽ arcsin r Ṽ -3 = 6 π E Ṽ arcsin r Ṽ ,
where the second line follows from the definition of Y; the third line follows from the law of iterated expectations and the fact that orthant probabilities depend only on correlations when the mean is equal to zero; the fourth line follows from computing the orthant probability of the normal, see, e.g., [START_REF] David | A note on the evaluation of the multivariate normal integral[END_REF]. Thus we can express Spearman's rho as an expectation, over random variable Ṽ, of a bivariate normal orthant probability, whose correlation is the only part that depends on Ṽ. The expression of the mixing density is obtained by considering the trivariate transformation

U 1 = (W 1 + W 3 ) -1/2 , U 2 = (W 2 + W 3 ) -1/2 and Ṽ = W 3 {(W 1 + W 3 )(W 2 + W 3 )} -1/2 of the independent vector (W 1 , W 2 , W 3 ), with inverse transformation w 1 = (u 1 -u 2 ṽ)/(u 1 u 2 2 ), w 2 = (u 2 -u 1 ṽ)/(u 2 1 u 2 )
and w 3 = ṽ/(u 1 u 2 ), with Jacobian determinant 4/ (u 1 u 2 ) 4 , and integrating out u 1 and u 2 in the range u 1 ≥ 0 and u 1 ṽ ≤ u 2 ≤ u 1 /ṽ, which corresponds to w i > 0 for i ∈ {1, 2, 3}. Equation [START_REF] Abdous | Dependence properties of meta-elliptical distributions[END_REF] shows that the Spearman rank correlation of a scale mixture of normals is an expectation of a correlation mixture of the Spearman rank correlation of the Gaussian, with the mixing density given in (2).

Corollary 1 (Upper bound of Spearman's rho for scale mixtures of normals). Within the class of scale mixtures of normals (SMN), the normal has the largest Spearman rank correlation in absolute value for a given value of the correlation parameter r: |ρ

S MN | ≤ |ρ N |.
Proof. This follows from Proposition 1, Jensen's inequality and the correlation inequality. Without loss of generality, assume r > 0. Given that arcsin is increasing, odd and convex on [0, 1], we get

ρ S MN = (6/π) E Ṽ arcsin r Ṽ ≤ (6/π) arcsin rE Ṽ Ṽ ≤ (6/π) arcsin (r/2) = ρ N ,
where the first inequality is due to Jensen's inequality and the last inequality follows from

E Ṽ = E Ṽ1 Ṽ2 = E Ṽ1 E Ṽ2 + corr Ṽ1 , Ṽ2 var Ṽ1 var Ṽ2 1/2 ≤ E Ṽ1 2 + var Ṽ1 = E Ṽ2 1 = E {W 3 /(W 1 + W 3 )} = 1/2, with Ṽi = {W 3 /(W i + W 3 )} 1/2 , for i ∈ {1, 2}.
The inequality follows from the correlation inequality corr Ṽ1 , Ṽ2 ≤ 1 and the fact that Ṽ1 and Ṽ2 are identically distributed, and the last equality follows from the fact that the W i s are independent and identically distributed.

Proposition 2 (Spearman's rho for the generalized hyperbolic distribution). The Spearman rank correlation of a bivariate generalized hyperbolic distribution X ∼ GH 2 (λ, χ, ψ, P) is given by ( 1), with mixing density

f Ṽ (ṽ) = 1 -ṽ2 3λ/2 ṽλ/2+1 1/2 ṽ/(ṽ 2 +1) t 2λ (1 + ṽ2 )t -ṽ -λ/2-1 (1 -4t 2 ) 1/2 (1 -ṽt) 3λ/2 K 3λ χψ(1-ṽ 2 )(1-ṽt) ṽ{(1+ṽ 2 )t-ṽ} 1/2 K λ ( √ χψ) 3 dt.
Proof. As the generalized hyperbolic distribution is a scale mixture of normals, by Proposition 1, Spearman's rho is given by [START_REF] Abdous | Dependence properties of meta-elliptical distributions[END_REF]. Upon substitution of the density of the GIG(λ, χ, ψ) into (2), we obtain

f Ṽ (ṽ) = (C/2) ṽλ-1 ∞ 0 u 1 /ṽ u 1 ṽ I(u 1 , u 2 )du 2 du 1 ,
where

C = (ψ/χ) 3λ 2 K λ √ χψ 3 , I(u 1 , u 2 ) = {(u 1 -u 2 ṽ) (u 2 -u 1 ṽ)} λ-1 (u 1 u 2 ) 4λ exp - h(u 1 , u 2 ) 2 ,
and

h(u 1 , u 2 ) = ψ      1 u 2 1 + 1 u 2 2 - ṽ (u 1 u 2 )      + χ      u 1 u 2 ṽ + u 1 u 2 2 (u 1 -u 2 ṽ) + u 2 1 u 2 (u 2 -u 1 ṽ)      .
By symmetry of the domain of integration D = {u 1 ≥ 0, u 1 ṽ ≤ u 2 ≤ u 1 /ṽ} around the 45 degree line u 2 = u 1 and symmetry of the integrand

I(u 1 , u 2 ) = I(u 2 , u 1 ), f Ṽ (ṽ) = C ṽλ-1 ∞ 0 u 1 u 1 ṽ I(u 1 , u 2 )du 2 du 1 . Using the change of variable (u 1 , u 2 ) = 1/ R 1/2 cos(θ) , 1/ R 1/2 sin(θ) with Jacobian determinant -1/ 2 {R cos(θ) sin(θ)} 2 , we obtain f Ṽ (ṽ) = (C/2) ṽλ-1 π/4 arctan(ṽ) (cos(θ) sin(θ)) 2λ (1 + ṽ2 ) cos(θ) sin(θ) -ṽ λ-1 Ĩ(θ)dθ, with Ĩ(θ) = ∞ 0 R λ-1 exp - 1 2 χR -1 + ψR dR,
where λ = 3λ, χ = χ 1 -ṽ2 / ṽ (1 + ṽ2 ) cos(θ) sin(θ)ṽ , and ψ = ψ {1ṽ cos(θ) sin(θ)}. By recognizing the density of the generalized hyperbolic, which integrates to one over the positive real line,

Ĩ(θ) = 2Kλ χ ψ χ-λ χ ψ λ .
Collecting terms and using the final change of variable t = cos(θ) sin(θ) = sin(2θ)/2 with dt = cos(2θ)dθ and Jacobian determinant 1 -4t 2 -1/2 yields the result.

Remark 1. The Spearman rank correlation of the GH depends on χ and ψ only through their product. Thus, for example, when ψ = 0, regardless of the value of χ, the Spearman rank correlation of the Pearson VII is equal to that of the Student t with degrees of freedom ν. This can be seen also from the fact that W ∼ IG(ν/2, χ/2) = (χ/ν)IG(ν/2, ν/2). This leads to the same distribution for Ṽ, since χ/ν simplifies, and therefore the Spearman rank correlation of the Pearson of type VII is the same as that of the Student t.

Proposition 3 (Spearman's rho for the Bessel distribution). The Spearman rank correlation of the bivariate Bessel distribution X ∼ B 2 (q, P) is given by ( 1), with mixing density

f Ṽ (ṽ) = 4 Γ(3 + 3q) Γ(1 + q) 3 ṽq 1/2 ṽ/(ṽ 2 +1) t 2(1+q) {(1 + ṽ2 )t -ṽ} q (1 -4t 2 ) 1/2 (1 -ṽt) 3(1+q) dt.
Proof. As the Bessel distribution is a scale mixture of normals, by Proposition 1, Spearman's rho is given by [START_REF] Abdous | Dependence properties of meta-elliptical distributions[END_REF]. Upon substitution of the density of the Γ(q + 1, q + 1) into (2), the density of Ṽ follows from using polar coordinates with radius R and angle θ in the integral of (2), recognizing that the integral involving the radius is proportional to an absolute moment of the normal of order 5 + 6q, and further using the change of variable t = sin(2θ)/2. Alternatively, it can be obtained as a limit case of the mixing density of the generalized hyperbolic when χ = 0 by writing the two Bessel functions K ν (y) when y is small as Γ(ν)2 ν-1 y -ν as in [START_REF] Barndorff-Nielsen | Hyperbolic distributions and distributions on hyperbolae[END_REF], using the substitution λ = q + 1 and simplifying.

Remark 2 (Spearman's rho for the Laplace distribution). The Spearman rank correlation of the bivariate symmetric Laplace distribution X ∼ L 2 (P), with correlation matrix P is given by ( 1), with mixing density

f Ṽ (ṽ) = 4ṽ 4 -ṽ2 2           7 -8ṽ 2 + ṽ4 + 2(2 + ṽ2 ) ṽ(4 -ṽ2 ) 1/2 arctan          1 -ṽ2 4 -ṽ2 1/2 3 -ṽ2 ṽ                    .
This expression can be found by setting q = 0 in the mixing density of the Bessel and solving the integral.

Proposition 4 (Spearman's rho for the Student t). The Spearman rank correlation, ρ T , of a bivariate Student t distribution X ∼ T 2 (P, ν) is given by [START_REF] Abdous | Dependence properties of meta-elliptical distributions[END_REF], with mixing density

f Ṽ (ṽ) = 2 Γ(ν) 2 Γ(3ν/2) Γ(ν/2) 3 Γ(2ν) ṽν-1 (1 -ṽ2 ) ν/2-1 2 F 1 (ν, ν; 2ν, 1 -ṽ2 ), 0 < ṽ < 1, ( 3
)
where 2 F 1 (a, b; c, z) = Γ(c)/ {Γ(a)Γ(c -a)} 1 0 v a-1 (1 -v) c-a-1 (1 -vz) -b
dv is the Gauss hypergeometric function. Proof. As the Student t distribution is a scale mixture of normals, by Proposition 1, Spearman's rho is given by (1). To compute the density of Ṽ, we write Ṽ = Ṽ1 Ṽ2 , where, for i ∈ {1, 2},

Ṽi = {W 3 /(W i + W 3 )} 1/2 ,
and W i ∼ IG(ν/2, ν/2). We can also write

Ṽi = {(1/W i )/(1/W i + 1/W 3 )} 1/2 ,
where 1/W i ∼ Γ(ν/2, ν/2) and therefore ( Ṽ1 , Ṽ2 ) are square roots of the bivariate beta distribution defined in [START_REF] Olkin | A bivariate beta distribution[END_REF], and the density of Ṽ is obtained by further appealing to Theorem 2.1 of [START_REF] Nagar | Product and quotient of correlated beta variables[END_REF], who derive the distribution of the product of the components of the bivariate Beta distribution as

f t (t) = Γ(a + b + c)Γ(a + c)Γ(b + c) Γ(a)Γ(b)Γ(c)Γ(a + b + 2c) t a-1 (1 -t) c-1 2 F 1 (a + c, a + c; a + b + 2c, 1 -t),
where in our case a = b = c = ν/2. Finally, (3) is obtained from f t (t) through Ṽ = T 1/2 , with inverse transformation t = ṽ2 and Jacobian determinant 2ṽ.

Corollary 2 (Spearman's rho for Student t when the degrees of freedom ν → 0). In the limit when the degrees of freedom parameter of the Student t tends to zero, Spearman's rho for the Student t tends to the Kendall rank correlation of the normal:

lim ν→0 ρ T (r, ν) = τ N (r).
Proof. The moments of Ṽ are obtained from those of the product of the components of a bivariate Beta distribution in [START_REF] Nagar | Product and quotient of correlated beta variables[END_REF]:

E Ṽh = Γ(ν) 2 Γ(3ν/2)Γ {(ν + h)/2} Γ(2ν)Γ(ν/2) 2 Γ(ν + h/2)
3 F 2 (ν, ν, ν/2; 2ν, ν + h/2; 1), where 3 F 2 (a, b, c; d, e; z) is the generalized hypergeometric function, see, e.g., [START_REF] Luke | The Special Functions and Their Approximations[END_REF]. For any h > 0, taking the limit of these moments when ν → 0,

lim ν→0 E Ṽh = 1 3 , since lim ν→0 Γ(ν) 2 Γ(3ν/2)Γ((ν+h)/2) Γ(2ν)Γ(ν/2) 2 Γ(ν+h/2) = 1
3 and lim ν→0 3 F 2 (ν, ν, ν/2; 2ν, ν + h/2; 1) = 1. A power series expansion of arcsin(r Ṽ) in the expression of Spearman's rho for the Student t yields

ρ T (r, ν) = 6 π E Ṽ arcsin r Ṽ = 6 π rE( Ṽ) + 1 2 r 3 3 E( Ṽ3 ) + 1 2 3 4 r 5 5 E( Ṽ5 ) + • • • .
Using the limit of the moments of Ṽ, and recognizing the series expansion of arcsin(r), lim ν→0 ρ T (r, ν) = 2 π arcsin(r) = τ N (r).

We have shown that when the degrees of freedom approach zero, all moments of Ṽ are 1/3. This happens if the density collapses to a Bernoulli with probability 1/3. If instead we consider the limit when ν → ∞, the density of Ṽ has point mass at 1 2 . This follows by expressing

W i ∼ IG(ν/2, ν/2) as W i = (Q/ν) -1 , with Q ∼ χ 2 [ν]
. By the law of large numbers, when ν → ∞, Q/ν p -→ 1, as Q can be viewed as a sum of ν independent χ 2 [START_REF] Abdous | Dependence properties of meta-elliptical distributions[END_REF] variables. By the continuous mapping theorem, W i p -→ 1 and Ṽ = g(W 1 , W 2 , W 3 ) p -→ 1/2 since g(x, y, z) = z/ (x + z)(y + z) is a continuous function. For any h > 0, taking the limit when ν → ∞, the moments of Ṽ become powers of 1/2:

lim ν→∞ E Ṽh = 1 2 h ,
and the Spearman rank correlation of the Student t is equal to that of the Gaussian.

Remark 3 (Spearman's rho for Cauchy distribution). Whenever ν ≤ 2, the Student t has no second order moments and therefore correlation does not exist. However, Spearman's rho can still be computed. For instance when ν = 1, X follows a Cauchy distibution, whose Spearman rank correlation is given by (1), with mixing density

f Ṽ (ṽ) = - 2 π ln(ṽ) (1 -ṽ2 ) 3/2 ,
and when ν = 2, the mixing density is f Ṽ (ṽ) = 8ṽ (ṽ 2 -1) 3 {(ṽ 2 + 1) ln(ṽ) + 1 -ṽ2 }.

Fig. 1 shows the mixing densities f Ṽ (ṽ) for the Cauchy, the Student t with degrees of freedom ν = 2 and ν = 4, the Bessel with q = 3, the Laplace and the generalized hyperbolic with parameters (λ, √ χψ) = (-1, 2) and (λ, √ χψ) = (0, 0.01), and Table 1 shows the corresponding values of Spearman's rho for a range of values of the correlation parameter r. The Student t with low degrees of freedom, the Laplace, the Bessel and the generalized hyperbolic with λ = -1 exhibit only slightly lower vales of Spearman's rho than the normal. On the other hand, the Cauchy and the generalized hyperbolic with (λ, √ χψ) = (0, 0.01), whose mixing densities have tails with a vertical asymptote at 0 and 1, exhibit values of Spearman's rho that are much farther below those of the normal. Table 1: Spearman rank correlation for different values of the correlation parameter r and for different scale mixture of normal distributions: the normal (second column), the Cauchy (third column), the Student t distribution with ν = 2 and ν = 4 degrees of freedom (fourth and fifth columns), the Laplace (sixth column), the Bessel (seventh column) and the generalized hyperbolic distribution with parameters (λ, √ χψ) = (-1, 2) and (λ, √ χψ) = (0, 0.01) (eighth and ninth columns).

r Normal Cauchy Student t Laplace Bessel Generalized hyperbolic ν = 2 ν = 4 q = 3 λ = -1 λ = 0 √ χψ = 2 √ χψ = 0.

A rank-based estimation of the parameters of the Student t

Combined with the expression for Kendall's tau, our expressions for Spearman's rho can be used as a basis for a rank-based estimation method of the correlation and shape parameters of the Student t distribution. Such a method is copula-based, as the only information that it retains from the marginal distributions are the ranks, and rank correlations depend on the copula and not on the marginal densities (see, e.g., [START_REF] Nelsen | An Introduction to Copulas[END_REF]). Such a method is especially useful when the degrees of freedom are low. In that case, the method of moments only delivers consistent estimates when the degrees of freedom are larger than 4, which guarantees that kurtosis exists, and moreover, the estimators have finite variance only when the degrees of freedom are larger than 8 (see [START_REF] Resek | Estimation of the parameters of a general student's t distribution[END_REF]). Maximum likelihood estimation (MLE) on the other hand suffers from numerical problems and the existence of many local maxima (see [START_REF] Liu | ML estimation of the t distribution using EM and its extensions, ECM and ECME[END_REF]). Given that it is based on the ranks of the data and not on their absolute value, the rank-based method should be robust to outliers (see [START_REF] Croux | Influence functions of the Spearman and Kendall correlation measures[END_REF][START_REF] Xu | A comparative analysis of Spearman's rho and Kendall's tau in normal and contaminated normal models[END_REF]), which also makes it a good candidate starting value for MLE.

The idea of estimating copula parameters from rank correlations is not new. Estimation of the parameter of the Gaussian copula from Spearman's rank correlation was proposed in [START_REF] Iman | A distribution-free approach to inducing rank correlation among input variables[END_REF]. Moment estimation of the correlation parameters of the Student t copula by inversion of Kendall's tau, was discussed in [START_REF] Mcneil | Quantitative Risk Management: Concepts, Techniques and Tools[END_REF], but whereas we propose to estimate the degrees of freedom based on Spearman's rho, they use MLE to estimate the degrees of freedom. In [START_REF] Genest | Statistical inference procedures for bivariate Archimedean copulas[END_REF] the authors discuss what they call a "naive" method for the Archimedean copulas, which consists in estimating the parameter of the copula by inverting Kendall's tau. They trace the idea of inverting rank correlations to estimate copula parameters to [START_REF] Genest | Frank's family of bivariate distributions[END_REF] for the Frank and [START_REF] Oakes | Semiparametric inference in a model for association in bivariate survival data[END_REF] for the Clayton.

The rank-based method is based on inversion of the following transformation Υ from parameters (r, ν) to rank correlations (τ, ρ T ):

τ ρ T = Υ(r, ν) = 2 π arcsin(r) 6 π E Ṽ arcsin r Ṽ , (4)
in which Kendall's tau depends on the correlation parameter r according to τ = (π/2) arcsin(r), while Spearman's rho depends both on the correlation parameter r the degrees of freedom ν, through (1), along with the mixing density given in (3). The rank-based estimation procedure works provided that the inverse of Υ exists, which is true as long as the Jacobian is invertible. Define J, the Jacobian of the transformation Υ, whose determinant is

|J| = | ∂τ ∂r ∂ρ T ∂ν |, since ∂τ ∂ν =0. Since ∂τ ∂r =
(2/π)/(1r 2 ) 1/2 is always different from zero, the rank-based estimation procedure works, provided that ∂ρ T ∂ν 0. When r = 1 (r = -1), τ = 1 (τ = -1), and by Theorem 5.1.11 of [START_REF] Nelsen | An Introduction to Copulas[END_REF], this implies that ρ T = 1 (ρ T = -1). When r = 0, ρ T = 0, since arcsin(0) = 0. Thus in these three cases, Spearman's rho does not depend on ν, and therefore ν is not identifiable from ρ T .

Next, we establish consistency and asymptotic normality of the rank-based estimators of the correlation parameter r and the degrees of freedom ν under the following assumption, which guarantees that Υ -1 , the inverse of Υ defined in (4), exists.

Assumption 1. The Spearman rank correlation of the Student t, ρ T (r, ν) = 6 π E Ṽ arcsin r Ṽ is strictly increasing (decreasing) in the degrees of freedom ν when 0 < r < 1 (-1 < r < 0).

Consistency of rank-based estimation. Under Assumption 1, the rank-based estimators of the correlation parameter

and the degrees of freedom (r, ν) are consistent estimators whenever r ∈ (-1, 1) \ {0}.

This follows from the fact that, since they are U-statistics, the empirical versions of Kendall's tau, τ, and Spearman's rho, ρ, are consistent estimators of their population counterparts τ and ρ T (see [START_REF] Puri | On a class of rank order tests for independence in multivariate distributions[END_REF]). Assumption 1 implies that r ∂ρ T ∂ν > 0, which establishes that |J| > 0, and therefore that Υ -1 exists and is differentiable and therefore continuous, provided that r ∈ (-1, 1) \ {0}. The consistency of the rank-based estimators (r, ν) then follows from an application of the continuous mapping theorem.

Asymptotic normality of rank-based estimator. Under Assumption 1, whenever r ∈ (-1, 1) \ {0}, the rank-based estimators r and ν are asymptotically normally distributed:

√ n r -r ν -ν ∼ N 2 0, (J -1 ) ⊤ ΣJ -1 , with Σ = σ 2 τ σ τ,ρ σ τ,ρ σ 2 ρ ,
where expressions for σ 2 τ , σ 2 ρ and σ τ,ρ are given in [START_REF] Hoeffding | A class of statistics with asymptotically normal distribution[END_REF]. This is a consequence of the fact, shown in [START_REF] Hoeffding | A class of statistics with asymptotically normal distribution[END_REF], that Kendall's tau and Spearman's rho are U-statistics that follow a joint asymptotic normal distribution:

√ n τ -τ ρT -ρ T ∼ N 2 (0, Σ) .
Given Assumption 1, Υ -1 exists, and since ∂ i+ j Υ/∂r i ∂ν j exists for all i + j ≤ 2, Υ -1 is continuous. Under these conditions, the result follows by a standard application of the delta method. Assumption 1 is supported by extensive numerical simulations, see Fig. 2, which shows the values of Spearman's rho for the Gaussian for different values of r, and the corresponding values of Spearman's rho of the Student t for degrees of freedom ν ranging from 0 to 10. For positive values of r, as the degrees of freedom ν increase, the Spearman rank correlation of the Student t converges from below to the value of that of the Gaussian. Since arcsin is an odd function, when the correlation parameter r is negative, the sign of the Spearman rank correlation is reversed, and the convergence is from above. The assumption states that, for a given correlation parameter r, the degrees of freedom parameter ν acts like a dependence parameter. It implies that the tail dependence of the Student t for low degrees of freedom comes at the expense of lower overall dependence, as measured by Spearman's rho. Combined with Corollary 2, the assumption implies that the Spearman rank correlation of the Student t is always greater than Kendall's tau in absolute value. This holds, even though the Student t is neither left-tail decreasing nor right-tail increasing, which are sufficient conditions for Spearman's rho to be greater than Kendall's tau in absolute value (see [START_REF] Capéraà | Spearman's rho is larger than Kendall's tau for positively dependent random variables[END_REF][START_REF] Fredricks | On the relationship between Spearman's rho and Kendall's tau for pairs of continuous random variables[END_REF]). More specifically, the Spearman rank correlation of the Student t increases from the value of Kendall's tau to the value of the Spearman rank correlation of the Gaussian, as the degrees of freedom increase from zero to infinity.

We can also show that a sufficient condition for Assumption 1 is the following assumption about the moments of the distribution of Ṽ, which is also supported by numerical evidence: Assumption 2. The odd moments of order h ≥ 3 of Ṽ, E Ṽh are strictly decreasing in the degrees of freedom ν.

To see that Assumption 2 implies Assumption 1, we use a power series expansion of arcsin(r Ṽ) in ρ T = 6 π E Ṽ arcsin r Ṽ , and setting r = 1, ρ T = 1, which yields

π 6 = E Ṽ + 1 2 E Ṽ3 3 + 1 2 3 4 E Ṽ5 5 + • • •
Taking derivatives with respect to the degrees of freedom ν,

∂E Ṽ ∂ν + 1 2 1 3 ∂E Ṽ3 ∂ν + 1 2 3 4 1 5 ∂E Ṽ5 ∂ν + • • • = 0.
Combining this last equation with the expression of the derivative of Spearman's rho with respect to the degrees of freedom ν, ∂ρ

T (r, ν) ∂ν = 6r π         1 2 1 3 (r 2 -1) ∂E Ṽ3 ∂ν + 1 2 3 4 1 5 (r 4 -1) ∂E Ṽ5 ∂ν + • • •         ,
which is positive (negative), whenever r > 0 (r < 0), provided Assumption 2 holds. Fig. 3 shows the first four odd moments of Ṽ across a range of values of the degrees of freedom ν. As can be seen in the figure, in the limit when the degrees of freedom ν → 0, E Ṽh = 1/3 for all h ≥ 0, whereas, when ν → ∞, the distribution collapses to a point mass at 1/2, and therefore E Ṽh = (1/2) h . In order to check how the rank-based procedure works in practice, we conduct a Monte Carlo simulation exercise with 1000 replications and for samples of size 1000 and 10000. Panel A of Table 2 shows the average bias from the rank-based estimation of the correlation parameter r, as well as the degrees of freedom ν of the Student t for the sample size of 1000. Correlation parameters are well-estimated for all possible cases of the Student t. The degrees of freedom parameter of the Student t shows a slight positive bias for small correlations (r = 0.1), which decreases for correlation parameters of about 0.2 or 0.3 and seems to flatten out for higher values of r. Although the absolute bias tends to increase with the degrees of freedom, this trend is much less pronounced for relative bias. Panel B shows that when the sample size increases, the bias vanishes for all values of the parameters. 0.1 0.0030 0.0002 0.0109 -0.0009 0.0328 -0.0003 0.2 0.0008 -0.0003 0.0031 0.0001 0.0027 0.0003 0.3 -0.0008 0.0002 0.0032 -0.0002 0.0055 -0.0000 0.4 -0.0017 -0.0004 0.0066 0.0002 0.0160 0.0002 0.5 0.0017 -0.0006 0.0013 -0.0006 -0.0036 -0.0002 0.6 0.0001 -0.0002 -0.0017 -0.0002 0.0150 -0.0001 0.7 0.0016 0.0003 0.0014 -0.0001 -0.0012 0.0002 0.8 0.0009 0.0000 0.0019 -0.0000 0.0087 -0.0002 0.9 0.0015 -0.0000 -0.0028 -0.0001 0.0167 -0.0001 We apply our procedure to five years of daily log-return data for Intel (INTC), Microsoft (MSFT) and General Electric (GE) from 1996 to 2000. This is the same data as in Example 5.57 of [START_REF] Mcneil | Quantitative Risk Management: Concepts, Techniques and Tools[END_REF]. We estimate the correlation and degrees of freedom parameters for each pair of returns using the rank-based estimation procedure and we obtain (r, ν) = (0.594, 8.143), (0.355, 5.110), (0.416, 7.222), respectively for the INTC-MSFT, INTC-GE and MSFT-GE pairs. As a comparison, in Example 5.59, [START_REF] Mcneil | Quantitative Risk Management: Concepts, Techniques and Tools[END_REF] obtain correlation parameters of (0.59, 0.36, 0.42) and degrees of freedom of 6.5 in a MLE estimation of a trivariate Student t copula from the ranks of the same data. The correlation parameters are extremely similar across both procedure, and their degrees of freedom are close to the average of the degrees of freedom over the three pairs (6.825).

Fig. 1 :

 1 Fig.1: This figure shows the mixing density f Ṽ ( Ṽ) for the Cauchy (plus), the Student t with ν = 2 degrees of freedom (square), the Student t with ν = 4 degrees of freedom (dotted), the Laplace (dash-dotted), the Bessel with q = 3 (circles) and the generalized hyperbolic with (λ, √ χψ) = (-1, 2) (solid) and (λ, √ χψ) = (0, 0.01) (dashed), respectively.

Fig. 2 :

 2 Fig. 2: This figure shows as solid horizontal lines the values of Spearman's rho of the Gaussian for values of the correlation parameter r of 0.25, 0.5 and 0.75, and as dotted lines the corresponding values of Spearman's rho for the Student t with degrees of freedom ν between 0 and 10.

MomentsFig. 3 :

 3 Fig.3: This figure shows the first four odd moments of ν, E Ṽh for h ∈ {1, 3, 5, 7}, when the degrees of freedom ν vary between 0 and 10. The solid line represents the first moment (h = 1), which increases to 0.5 as ν increases, while the dashed lines represent the third, fifth and seventh moment (h ∈ {3, 5, 7}), which decrease to 0.5 h as ν increases.

Table 2 :

 2 Bias of the rank-based estimation of the parameters of the bivariate Student t distribution for different values of the correlation parameter r, based on a Monte Carlo simulation with 1000 replications and a sample size of 1000 in panel A and 10000 in panel B. We consider the Student t distribution with degrees of freedom ν = 1, ν = 2 and ν = 4. For each distribution we show the bias in the shape parameter, followed by the bias in the correlation parameter. Student t Panel A: sample size 1000 r ν = 1 r ν = 2 r ν = 4 r 0.1 0.1176 -0.0011 0.3831 -0.0035 0.6849 0.0007 0.2 0.0136 -0.0002 0.0697 -0.0014 0.1559 -0.0010 0.3 0.0085 -0.0008 0.0118 -0.0003 0

Acknowledgments

This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01). The authors also acknowledge the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-17-CE26-0001 (project BREAKRISK). We thank William Kengne for helpful discussions, and the editor as well as two anonymous referees whose suggestions greatly helped improve the readability of this paper.


Thierry Huillet

RANDOM WALKS PERTAINING TO A CLASS OF DETERMINISTIC WEIGHTED GRAPHS

In this note, we try to analyze and clarify the intriguing interplay between some counting problems related to specific thermalized weighted graphs and random walks consistent with such graphs.

Introduction

The purpose of this work is to underline the subtle relationship between some counting problems related to thermalized weighted graphs and random walks consistent with such graphs; see e.g. [START_REF] Aldous | Reversible Markov Chains and Random Walks on Graphs[END_REF] for a detailed treatise on general random walks on graphs. Let us summarize the topics developed in this paper. We start with defining finite thermalized weighted graphs. We show that the notion of a graph singularity spectrum naturally arises in the problem consisting in counting the number of paths whose transition-energy rate is asymptotically of a given order. This spectrum is classically the Legendre transform of the graph pressure function which is the logarithm of the spectral radius of its weight matrix. The corresponding Perron eigenvectors play a key role in the Perron-Frobenius theory. We next recall the Gibbs-variational principle, stating that the pressure produced by all random walks consistent with the graph structure is bounded below by the graph pressure which may itself be viewed as the pressure of some consistent canonical random walk. This turns out to be a by-product of the Ruelle thermodynamic formalism. We then exhibit and interpret some important consistent random walks in the light of quasi-stationary distributions for substochastic random walks. The idea is to normalize the weight matrix of the graph by its norm to make it substochastic so that, by adding an extra absorbing coffin state, we may switch to the study of a proper random walk conditioned to its absorption time. By doing so, a probabilistic interpretation of both the spectral radius and Perron eigenvectors of the graph weight matrix naturally comes out and at least two conditioning are shown to be relevant: one is to condition locally the above random walk on not hitting the absorbing state in one step at each iteration, the other is to condition it on not hitting the absorbing state in the remote future. The latter construction is shown to be the canonical random walk with smallest pressure production rate. At the end of the paper, we briefly discuss three particular cases, namely: the case where the weighted graph is reduced to its adjacency matrix, the case of a potential weighted graph and the case of a symmetric reversible weighted graph. Several general conclusions that make use of the above constructions may be drawn. One is an expression in terms of the average transition energy of the canonical random walk associated to the adjacency matrix of the value at which the singularity spectrum of any weighted graph attains its maximum, another is that the entropy production rate of the locally conditioned random walk is always bounded above by the logarithm of the spectral radius of its adjacency matrix, in the potential case.

Finite graphs with Boltzmann weights

Let W ≥ 0 be some non-negative N × N weight matrix of some finite graph (i.e. with non-negative entries W (i, j) ≥ 0). Let A = [A (i, j)], defined by the indicator function:

A (i, j) = I (W (i, j) > 0) ∈ {0, 1} ,
stand for the Boolean adjacency matrix associated to W . With A ′ denoting the transpose of A, we shall assume that A = A ′ and that A is irreducible: in other words, the underlying topological graph is undirected and strongly connected so that for each couple of states (i, j), there is an integer m such that A m (i, j) > 0.

With β ∈ R, we shall choose to represent W under the form:

(1)

W (i, j) =: W β (i, j) = A (i, j) e -βH(i,j) ,
for some well-behaved transition energies -∞ < H (i, j) < +∞ from state i to j, not all equal to the same value. The matrix W β therefore appears to be the weight matrix of some thermalized weighted graph: it can be represented as the Hadamard product (say, * ) of A with some positive Boltzmann kernel matrix K β with entries K β (i, j) = e -βH(i,j) :

(2)

W β = A * K β .
We note that the Hadamard λ-power

(λ > 0) of W β simply is W * λ β = W λβ = A * K λβ ,
corresponding to a rescaling of β.

Remark: Let x 1 < x 2 < ... < x N be N points on the line (circle). For some matrix H, we may define

H (i, j) = H (x i , x j )
to be the interaction energy between sites (i, j) in positions (x i , x j ) leading to a slightly more general spatially extended model that can be treated along similar lines. △ 2.1. Some counting problems arising in this context. The quantity

W β (i n ) := n m=1 W β (i m-1 , i m )
is the weight of the n-path i n := {i 0 , i 1 , .., i n } which is non-null if and only if:

A (i n ) := n m=1 A (i m-1 , i m ) = 0.
The total product weight of n-step paths connecting states (i 0 , i n ) therefore is given by the corresponding element of the transfer matrix:

W n β (i 0 , i n ) = i1,..,in-1 n m=1 W β (i m-1 , i m ) = i1,..,in-1 n m=1 A (i m-1 , i m ) e -β P n m=1 H(im-1,im) = Nn(i0,in) p=1 e -βHn(p) ,
where H n (p) is the cumulative energy of the p-th path connecting (i 0 , i n ) and N n (i 0 , i n ) := A n (i 0 , i n ) is the number of such n-paths. Summing over the endpoints of the n-paths, we obtain the full partition function of energy

(3) Z n (β) := i0,in W n β (i 0 , i n ) = i0,..,in n m=1 W β (i m-1 , i m ) = Nn p=1 e -βHn(p) ,
where

N n := i0,in N n (i 0 , i n ) = Z n (0) is the total number of n-paths. Define N n,ε (α) := # p ∈ [N n ] : 1 n H n (p) ∈ (α -ε, α + ε) ,
the number of n-paths whose transition-energy rate is asymptotically of order α.

We expect (4) lim

ε↓0 lim n↑∞ 1 n log N n,ε (α) = f (α) ≥ 0, where f (α) = inf β (αβ -p (β)), α ∈ [α -, α + ],
is the concave Legendre transform of some concave pressure function p (β). We may call f (α) the singularity spectrum of the weighted graph.

Observing from (3) that Z n (β) = W n β is a matrix-norm and recalling under our irreducibility assumption:

W n β 1/n → n↑∞ ρ β , the spectral radius of W β , we get (5) - 1 n log Z n (β) → n↑∞ p (β) = -log ρ β .
For each β, the number ρ β , as an eigenvalue of W β , satisfies det 1

ρ β W β -I = 0.
For each n, the quantity

Φ n (β) := Z n (β) Z n (0) = 1 N n Nn p=1 e -βHn(p)
is the Laplace-Stieltjes transform of some discrete probability measure on n-paths satisfying -

1 n log Φ n (β) → p (β) -p (0) = -log ρ β ρ 0
. This limit therefore is the log-Laplace transform of some probability distribution which, in particular, is smooth and concave. The pressure p (β) is classically related to the scaled free energy τ (β) by: p (β) =: βτ (β) . Note that, for each n, we also have:

Φ n (β) = 1 N n hn∈Hn N n (h n ) e -βhn ,
where

H n := Span(H n (p) ; p ∈ [N n ]) , N n (h n ) := # {p ∈ [N n ] : H n (p) = h n } and N n (h n ) /N n is the probability of n-paths of energy h n .
With 1 = (1, 1, .., 1) ′ , we define w β := W β 1 to be the column-sum vector of W β , with entries w β (i) = j A (i, j) e -βH(i,j) . Define w

+ β = max i w β (i), w - β = min i w β (i). We have w - β ≤ ρ β ≤ w + β and so -log w + β ≤ p (β) ≤ -log w - β .
Clearly, it holds that

-log w + β ∼
β↑∞ βα -where α -= min i,j:A(i,j)=1 H (i, j) and similarly,

-log w - β ∼ β↓-∞ βα + where α + = max i,j:A(i,j)=1 H (i, j) > α -.
We can check that f (α -) = f (α + ) = 0 and that the maximum of f (α) is attained at α = α 0 = p ′ (0). We also have

f (α) = f (p ′ (β)) = p ′ (β) β -p (β) so that f (α 0 ) = -p (0) = log ρ 0 > 0 where ρ 0 is the spectral radius of A.
Whenever α -< 0, α + > 0, then there is a β c , possibly not equal to 0, given by:

p ′ (β c ) = 0. With α c := p ′ (β c ) = 0, f (α c ) = -p (β c ) > 0.
For all distinct pairs of nodes (i, j), let m (i,

j) = inf (m > 1 : A m (i, j) > 0) . Then m * = max (i,j) m (i, j)
is the diameter of the adjacency graph. For each (i, j), there can be more than one path of minimal length m (i, j) . Let N m(i,j) ≥ 1 be the number of such length-m (i, j) paths and let h (i, j) be the energy of any path with smallest energy among these N m(i,j) paths. Then

α * = max (i,j) [h (i, j) /m (i, j)]
is a quantity of interest related to the energy diameter of the weighted graph.

Clearly, α -< α * < α + and α * belongs to the range of the spectrum.

2.2.

Perron-Frobenius and the like. Let π ′ β > 0 and ϕ β > 0 be the line and column (l 1 -norm 1) Perron vectors of W β associated to the spectral radius ρ β of W β :

(6) ρ β π ′ β = π ′ β W β and ρ β ϕ β = W β ϕ β .
Under our hypothesis, ρ β > 0 is the algebraically simple real dominant eigenvalue of W β . If A is in addition primitive (A m > 0 for some integer m), then all the other eigenvalues of W β are strictly contained within the disk: |ρ| < ρ β , else some could lie on the disk |ρ| = ρ β because of the underlying periodicity of the problem.

We shall let φ β := ϕ β / π ′ β ϕ β in such a way that the Hadamard product of π β and φ β , namely the column vector

π β * φ β , with components π β * φ β (i) = π β (i) φ β (i), has l 1 -norm 1 (i.e. π ′ β φ β = 1). Remarks: (i) When H (i, j) = H (j, i), W β is itself symmetric (W β = W ′ β ) then π β = ϕ β and π β * φ β = ϕ β * ϕ β / ϕ ′ β ϕ β .
Then it is useful to introduce the probability wave vector of l 2 -norm 1 :

ψ β = ϕ β / ϕ ′ β ϕ β 1/2
in such a way that

π β * φ β = ψ β * ψ β .
(ii) Letting ε β := 1ρ β /w + β ≥ 0 stand for the scaled spectral gap of the graph, the equation giving the right eigenvector ϕ β may be recast as

ε β ϕ β = I - 1 w + β W β ϕ β =: -∆ β ϕ β , where ∆ β = 1 w + β W β -I is a Laplacian of the graph. Observe that w + β = |W β | ∞ is the matrix norm induced by the l ∞ -vector norm so that ρ β /w + β ≤ 1 (i.e. ε β ≥ 0) and that log ρ β / log w + β → β↑∞ 1. ▽
Consistently with (6), we shall let π ′ 0 > 0 and ϕ 0 > 0 stand for the line and column Perron vectors of A = W 0 with:

ρ 0 π ′ 0 = π ′ 0 A and ρ 0 ϕ 0 = Aϕ 0 , associated to the spectral radius ρ 0 of A = W 0 . We shall let φ 0 := ϕ 0 / (π ′ 0 ϕ 0 ) so that the Hadamard product π 0 * φ 0 has l 1 -norm 1. Note that, since A = A ′ , π 0 = ϕ 0 and π 0 * φ 0 = ϕ 0 * ϕ 0 / (ϕ ′ 0 ϕ 0 ) =: ψ 0 * ψ 0 where ψ 0 = ϕ 0 / (ϕ ′ 0 ϕ 0 ) 1/2
is the wave vector associated to A.

Random walks on graphs

In the study of weighted graphs, questions pertaining to counting are then relevant. Once such weighted graphs have been introduced, it is useful to consider the following particular class of random walks attached to such graphs. Let 0 ≤ Π = [Π (i, j)] denote some stochastic matrix with column sums one: Π1 = 1. Let P W β be the set of stochastic matrices which are W β -consistent in the sense that [START_REF] Lambert | Population dynamics and random genealogies[END_REF] Π ∈ P W β ⇔ {Π (i, j) = 0 whenever W β (i, j) = 0} .

Let µ ′ > 0 be the line left Perron eigenvector of Π, satisfying: µ ′ = µ ′ Π (the unique invariant probability measure associated to Π). Clearly, to each such Π a positive recurrent random walk (RW, say) can be associated.

3.1.

A variational principle and first consequences. In this context, the following Gibbs-variational principle indeed holds [START_REF] Arnold | Evolutionary formalism for products of positive random matrices[END_REF], resulting from the Ruelle thermodynamic formalism [START_REF] Ruelle | Thermodynamic formalism. The mathematical structures of equilibrium statistical mechanics[END_REF]. It reads:

log ρ β = sup Π∈PW β   - i µ (i) j Π (i, j) log Π (i, j) + i µ (i) j Π (i, j) log W β (i, j)   (8)
= sup

Π∈PW β   - i µ (i) j Π (i, j) log Π (i, j) -β i µ (i) j Π (i, j) H (i, j)   ,
where the supremum is attained for the unique stochastic matrix Π * which is W β -consistent and defined by the Doob transform:

(9) Π * (i, j) = 1 ρ β W β (i, j) φ β (j) φ β (i) . With D φ β := diag φ β , this is also; Π * = 1 ρ β D -1 φ β W β D φ β , in matrix form.
The corresponding invariant measure satisfying µ ′ * = µ ′ * Π * can easily be checked to be: [START_REF] Ruelle | Thermodynamic formalism. The mathematical structures of equilibrium statistical mechanics[END_REF] µ * = π β * φ β , the Hadamard product of the left and right eigenvectors of the weight matrix W β .

We shall call the RW with transition probability Π * the canonical RW consistent with W β . Using this canonical RW construction, we get:

log ρ β = - 1 ρ β i π β (i) φ β (i) j W β (i, j) φ β (j) φ β (i) log 1 ρ β φ β (j) φ β (i) = - 1 ρ β i π β (i) j W β (i, j) φ β (j) -log ρ β + log φ β (j) -log φ β (i) = log ρ β + i π β (i) φ β (i) log φ β (i) - 1 ρ β i π β (i) j W β (i, j) φ β (j) log φ β (j) ,
leading to the expression

ρ β = i π β (i) j W β (i, j) φ β (j) log φ β (j) i π β (i) φ β (i) log φ β (i)
, in terms of the left and right Perron eigenvectors of W β . As a result, we obtain

(11) p (β) =: βτ (β) = -log i π β (i) j A (i, j) e -βH(i,j) φ β (j) log φ β (j) i π β (i) φ β (i) log φ β (i)
.

From ( 8), for all W β -consistent stochastic matrix Π = Π * :

(12) log ρ β > - i µ (i) j Π (i, j) log Π (i, j) -β i µ (i) j Π (i, j) H (i, j) .
In the right-hand-side of (12), s :=i µ (i) j Π (i, j) log Π (i, j) =: i µ (i) s (i) is the equilibrium Shannon entropy production rate of the ergodic Markov chain governed by Π and u := i µ (i) j Π (i, j) H (i, j) =: i µ (i) u (i) its equilibrium internal transition energy. It follows from (8) that the quantity p (β) :=log ρ β is a universal lower bound for the equilibrium pressure production rate of all W β -consistent walkers. Stated differently, defining the pressure of a consistent RW governed by Π ∈ P W β , Π = Π * , as:

p Π (β) := β i µ (i) j Π (i, j) H (i, j) + i µ (i) j Π (i, j) log Π (i, j) ,
it holds that:

(13) for all Π = Π * ∈ P W β , p Π (β) > p (β) = p Π * (β) .
Remark: On the other hand, we also recall the Friedland-Karlin inequality [START_REF] Friedland | Some inequalities for the spectral radius of non-negative matrices and applications[END_REF] of a similar flavor:

(14) p (β) = -log ρ β ≥ i π β (i) φ β (i) log w β (i)
where, w β := W β 1 is the column-sum vector of W β : w β (i) = j A (i, j) e -βH(i,j) . It gives a universal lower bound of p (β) in terms of the invariant measure µ

* (i) = π β (i) φ β (i) associated to Π * . △ 3.2.
The entropy production rate of the RW governed by Π ∈ P W β . We need to say a few words on the way to compute the quantity s associated to some Π. We refer to [START_REF] Rached | Rényi's divergence and entropy rates for finite alphabet Markov sources[END_REF] for additional information. For each pair of connecting states (i 0 , i n ) , define:

Π n λ (i 0 , i n ) := i1,..,in-1 n m=1 Π (i m-1 , i m ) λ ,
where Π (i m-1 , i m ) λ is the (i m-1 , i m ) entry of Π * λ , the Hadamard λ-power of Π, λ > 0. Define the Rényi λ-entropy of all n-paths of the RW governed by Π and started using the invariant measure µ to be

R n (λ) := 1 1 -λ log i0,in µ (i 0 ) Π n λ (i 0 , i n ) .
Note that with Π (i n ) := n m=1 Π (i m-1 , i m ) the probability of the n-path i n := {i 0 , i 1 , .., i n } :

R n (λ) → λ↑1 S n = - in µ (i 0 ) Π (i n ) log Π (i n ) ,
the Shannon entropy of n-paths at equilibrium. Then, with α (λ) :

= i µ (i) j Π (i, j) λ : (15) 1 n R n (λ) → n↑∞ r (λ) := 1 1 -λ log α (λ)
where r (λ) is the Rényi-entropy production rate of the walker. As a result,

r (λ) → λ↑1 s = - i µ (i) j Π (i, j) log Π (i, j) = -α ′ (1) .
This approach is useful to compute the Shannon-entropy production rate s for specific Πs.

3.3.

Random walks consistent with W β . We now exhibit and interpret some important W β -consistent RWs in the light of quasi-stationary distributions for substochastic RWs. By doing so, a probabilistic interpretation of ρ β , π β and φ β emerges.

Let us first normalize W β in the following way. Consider the matrix:

(16)

W β := W β W β ,
for some matrix-norm W β of W β . For example:

W β = N max i,j W β (i, j) or W β = i,j W β (i, j) or W β = w + β = max i w β (i).
The spectral radius of W β now is ρ β = ρ β / W β < 1 with the same left and right strictly positive Perron eigenvectors π β > 0 and ϕ β > 0 as for W β in [START_REF] Horn | Matrix analysis[END_REF]. By doing so, the matrix W β is substochastic in the sense that, with w β := W β 1 the columnsum vector of W β , then: w β (i) ∈ (0, 1] with w β (i) < 1 for at least one i. To recast this problem into a stochastic problem, we may add an additional coffin state, say ∂ := {0} and look at the enlarged (N + 1) × (N + 1) stochastic matrix P :

(17) P = 1 0 ′ 1-w β W β .
P now is the stochastic transition matrix of a RW, say {X n } , having state {0} as an additional absorbing state. Let τ 0 be the first hitting time of ∂ = {0} for this RW {X n }.

Using this construction, clearly, the substochastic matrix W β turns out to be the transition matrix of the process X n •I (τ 0 > n) (i.e. X n , restricted to the set τ 0 > n).

In other words, with e ′ i0 the line-vector with a single 1 in position i 0 , 0 elsewhere, we have

P i0 (X n = i n , τ 0 > n) = W n β (i 0 , i n ) = e ′ i0 W n β e in , i 0 , i n ∈ {1, ., N } . Therefore, (18) P i0 (τ 0 > n) = e ′ i0 W n β 1. We note that P i0 (τ 0 = 1) = P i0 (τ 0 > 0) -P i0 (τ 0 > 1) = e ′ i0 I -W β 1 = 1 - w β (i 0 ) , the probability mass defect of W β at state i 0 .
For all (i 0 , i n ), we have:

lim n↑∞ W n β (i 0 , i n ) 1/n
= ρ β and, only when A is primitive (irreducible and aperiodic), by the strong version of Perron-Frobenius theorem (see [START_REF] Horn | Matrix analysis[END_REF])

(19) lim n↑∞ ρ -n β W n β = φ β π ′ β ,
where π ′ β > 0 and φ β > 0, defined in [START_REF] Horn | Matrix analysis[END_REF], are the left-(right-) eigenvectors of W β associated to ρ β , chosen, as before, so as to satisfy π ′ β φ β = 1. As a result of ( 18) and ( 19), when A is primitive:

(20) lim n↑∞ ρ -n β P i0 (τ 0 > n) = φ β (i 0 ) ,
meaning that τ 0 is tail-equivalent to a geometric random variable with success probability ρ β . The latter formula therefore gives the limiting interpretation of φ β in the context of the RW {X n }. What about π β ?

Firstly, because π ′ β is the left eigenprobability vector of W β with eigenvalue ρ β :

(21)

P π β (τ 0 > n) := N i0=1 π β (i 0 ) P i0 (τ 0 > n) = π ′ β W n β 1 = ρ n β .
If the process is started with π β , the law of τ 0 is exactly geometrically distributed on {1, 2, ..., N } with success probability ρ β .

Consider now the conditional probability

P i0 (X n = i n | τ 0 > n). Recalling P i0 (X n = i n , τ 0 > n) = e ′ i0 W
n β e in , by Bayes rule, we get

P i0 (X n = i n | τ 0 > n) = e ′ i0 W n β e in e ′ i0 W n β 1 = e ′ i0 ρ -n β W n β e in e ′ i0 ρ -n β W n β 1 ,
showing that, independently of the starting point i 0

P i0 (X n = i n | τ 0 > n) → n↑∞ e ′ i0 φ β π ′ β e in e ′ i0 φ β π ′ β 1 = π β (i n ) .
Such a probability measure π β is called a Yaglom limit [START_REF] Yaglom | Certain limit theorems of the theory of branching random processes[END_REF] of {X n }.

Further, with

P π β (•) := N i=1 π β (i 0 ) P i0 (•), for each n, i n ∈ {1, 2, .., N } : (22) P π β (X n = i n | τ 0 > n) := P π β (X n = i n , τ 0 > n) P π β (τ 0 > n) = π ′ β W n β e in π ′ β W n β 1 = π β (i n ) ,
and this precisely means that π β is the (unique) quasi-stationary distribution (QSD) of {X n }. As is well-known for Markov chains with finite state-space absorbed at ∂, we observe that the Yaglom limit coincides with its QSD. When A is primitive (strongly connected and aperiodic), Equations ( 21), ( 22) and ( 20) provide a natural interpretation of ρ β , π β and φ β in terms of the RW governed by P in (17) and its stopping time τ 0 .

We refer to [START_REF] Lambert | Population dynamics and random genealogies[END_REF] for additional informations on QSD and Yaglom limits in the context of population dynamics.

Remark: When A is irreducible but not primitive (the underlying topological graph is strongly connected but periodic), only the following weaker form of the Perron-Frobenius theorem holds true [START_REF] Horn | Matrix analysis[END_REF] lim

K↑∞ 1 K K n=1 ρ -n β W n β = φ β π ′ β . With φ + β = max i φ β (i), φ - β = min i φ β (i), Equation (20) has to be weakened into φ - β /φ + β ≤ ρ -n β P i0 (τ 0 > n) ≤ φ + β /φ - β ,
for all n, i 0 . The quantity ρ -n β P i0 (τ 0 > n) may oscillate and not tend to some limit; however

-1 n log P i0 (τ 0 > n) → ρ β still holds true. △
The above construction of the RW {X n } allows now to interpret two fundamental W β -consistent RWs.

The locally conditioned random walk: With D w β := diag(w β ), the transition matrix of the one-step conditioned process (X 1 | τ 0 > 1) is:

Π (i, j) = e ′ i W β e j e ′ i W β 1 = e ′ i D -1 w β W β e j = e ′ i W β e j e ′ i W β 1 = e ′ i D -1 w β W β e j ,
normalizing each line i by e ′ i W β 1 = w β (i) . Clearly, Π1 = 1 and the RW with transition matrix :

(23) Π = D -1 w β W β is W β -consistent. Note that Π is invariant under the scaling W β → W β .
Let µ be the invariant associated to this Π. It holds, [START_REF] Romanovsky | Discrete Markov chains. Translated from the Russian by E. Seneta[END_REF], that

µ (i) = (I -Π) i,i i (I -Π) i,i
where (I -Π) i,i is the cofactor of the (i, i) -entry of the matrix I -Π. Then, if ← -Π is the transition matrix of the reversed (backward in time) chain of Π at equilibrium:

← - Π ′ = D µ ΠD -1 µ .
In general, ← -Π = Π and detailed balance may not hold.

Global conditioning and the canonical process. Consider now the proper Markov chain whose transition probabilities are obtained by the Doob transform:

Π * (i, j) = ρ -1 β φ β (j) φ β (i) W β (i, j) = ρ -1 β φ β (j) φ β (i) W β (i, j) , i, j ∈ {1, .., N } , satisfying Π * 1 = 1.
In matrix form:

(24) Π * = ρ -1 β D -1 φ β W β D φ β ,
and Π * is also invariant under the scaling W β → W β . An important property of this RW is the following: The probability Π * (i n ) :

= n m=1 Π * (i m-1 , i m ) of the n-path i n is Π * (i n ) = ρ -n β W β (i n ) n m=1 φ β (i m ) φ β (i m-1 ) = ρ -n β W β (i n ) φ β (i n ) φ β (i 0 ) .
For a bridge n-path for which Doob transforms have to do with conditioning a process on its lifetime. The Markov chain with one-step transition probability matrix Π * may be shown to be the one of the process whose one-step transition probability distribution is:

i 0 = i n , Π * (i n ) = ρ -n β W β (i n ) reduces, up
(26) Π * (i, j) = lim n↑∞ P i (X 1 = j | τ 0 > n) ,
corresponding to X n conditioned to never hit the coffin state ∂ = {0} in the distant future; see [START_REF] Lambert | Population dynamics and random genealogies[END_REF]. This process has a unique invariant measure given by µ * in (25).

Defining as before ← -Π * by: ← -

Π ′ * = D µ * Π * D -1
µ * , we can ask conditions under which detailed balance ← -

Π * = Π * holds. We have ← - Π ′ * = ρ -1 β D π β D φ β D -1 φ β W β D φ β D -1 φ β D -1 π β = ρ -1 β D π β W β D -1 π β so that ← - Π * = ρ -1 β D -1 π β W ′ β D π β showing that reversibility holds when W β = W ′ β since if this is the case: π β = φ β .
No extra state. We emphasize here that there are some alternative ways to force the substochastic problem into a stochastic one. Assume A (i, i) > 0 for each i, in which case A N -1 > 0 and A necessarily is primitive. Consider the stochastic matrix Π which is W β -consistent:

(27) Π = W β + D 1-w β ,
where D 1-w β := diag(1w β ), satisfying Π1 = 1. In that case, the mass defect vector 1-w β is transferred to the diagonal entries of W β to make it stochastic, without appealing to an extra coffin-state. Note that Π in (27) no longer is invariant under the scaling W β → W β and so this normalization is norm-dependent.

Special cases

1. The topological case. Assume β = 0. In this case, W 0 = A and i) . log ρ 0 > 0 interprets as the maximal entropy production rate of all Markov chains governed by such Πs. The RW with transition matrix Π 0 * is termed the maximal entropy random walk in [START_REF] Burda | Localization of maximal entropy random walk[END_REF]. Its invariant measure is µ * (i) = ψ 0 (i) 2 . When Π = D -1 a A, with a = A1, the invariant measure is µ (i) = a (i) / i a (i) , proportional to the node degrees. Then s = i a (i) log a (i) / i a (i) and log ρ 0 > s is an inequality first discussed in [START_REF] Burda | Localization of maximal entropy random walk[END_REF]. When disorder is present, the canonical RW associated to W 0 = A was also shown therein to exhibit localization properties.

log ρ 0 > s = - i µ (i) j Π (i, j) log Π (i, j) for all A-consistent matrix Π = Π 0 * with: Π 0 * (i, j) = 1 ρ 0 A (i, j) φ 0 (j) φ 0 (
Consider the general inequality (12) for all W β -consistent stochastic matrix Π = Π * =: (i) . Choosing for Π the above particular value: Π = Π 0 * , we obtain log ρ β > log ρ 0 -

1 ρ β W β (i, j) φ β (j) φ β
β i ψ 0 (i) 2 j Π 0 * (i, j) H (i, j) .
Therefore, the average transition energy

(28) α 0 := i ψ 0 (i) 2 j Π 0 * (i, j) H (i, j)
interprets as the slope at β = 0 of the graph pressure function β → p (β), namely:

α 0 = p ′ (0) . We have f (α 0 ) = -p (0) = log ρ 0 .
2. The potential case: Assume H (i, j) = U (j) -U (i) for some potential U attached to the nodes of the graph. In this case, the matrix K β defining W β is called a potential kernel. Firstly, in this case, it follows from (28) and the equilibrium property of µ * = ψ 0 * ψ 0 , Π 0 * that: (29)

α 0 := i ψ 0 (i) 2 j Π 0 * (i, j) [U (j) -U (i)] = 0.
We conclude that the singularity spectrum of all graph with potential kernel K β attains its maximum at α 0 = 0.

With v β the column-vector with entries v β (i) = exp -βU (i), we get:

(30)

W β := A * K β = D -1 v β AD v β .
We have

w β = W β 1 = D -1 v β Av β and D w β = D -1 v β D Av β .
Note that W β is diagonally similar to A so that the spectral radius of W β is ρ 0 , independently of β.

-Consider first the RW with transition matrix Π = D -1

w β W β = D -1 Av β AD v β .
Its invariant measure is characterized by: µ ′ = µ ′ Π. Recalling A = A ′ , we find µ ∝ D v β Av β , with normalized entries weighting output degree nodes with lowest U :

(31) µ (i) = j A (i, j) e -β(Ui+Uj ) / i,j
A (i, j) e -β(Ui+Uj ) .

This RW with transition matrix Π is reversible because

← - Π = D -1 µ Π ′ D µ = Π.
-Secondly, consider the canonical RW consistent with

W β = D -1 v β AD v β . The right eigenvector φ β of W β = D -1 v β AD v β is φ β = D -1 v β φ 0 .
It is associated to the eigenvalue ρ 0 . Thus the canonical RW has transition matrix Π * is given by: (32)

Π * = ρ -1 0 D -1 φ β W β D φ β = ρ -1 0 D -1 φ 0 AD φ 0 = Π 0 * .
Its invariant measure is π * = ψ 0 * ψ 0 . The canonical RW consistent with the potential weight matrix W β = D -1 v β AD v β always coincides with the canonical RW consistent with its adjacency matrix A governed by Π 0 * .

With Π = D -1 w β W β with entries Π (i, j) = A (i, j) e -β(U(j)-U(i)) / j A (i, j) e -β(U(j)-U(i)) , and with invariant measure µ (i) displayed in (31), for all β, we get log ρ 0 > - We conclude that for potential kernels K β , the entropy production rate of the RW with probability transition matrix Π = D -1 w β W β = D -1 w β [A * K β ] is always bounded above by log ρ 0 .

Remark: The W β -consistent RW with transition matrix Π = D -1 w β W β associated to the weight kernel W β = AD v β was also considered in [START_REF] Gomez-Gardenes | Entropy rate of diffusion processes on complex networks[END_REF]. For this model, the cost of a transition from i to j only depends on the terminal state, regardless of where one starts from. Although the latter is not in the potential class, its invariant measure is also given by (31). ▽ 3. The symmetric case. If H (i, j) = H (j, i), then W β = W ′ β itself. For example H (i, j) = |U (j) -U (i)| for some potential U attached to the nodes of the graph, or H (i, j) is some distance (ultrametric or not) between nodes i and j. In this case, for all β, the invariant measure µ * of the canonical RW governed by Π * =

1 ρ β D -1 φ β W β D φ β , is: (33) µ * = ψ β * ψ β
and the corresponding RW is reversible.

When W β = W ′ β , the invariant measure associated to Π = D -1 w β W β satisfying µ ′ = µ ′ Π is given by: µ (i) = w β (i)

i w β (i) = j A (i, j) e -βH(i,j) i,j A (i, j) e -βH(i,j) . We have ← -

Π ′ = D µ ΠD -1 µ = D w β D -1 w β W β D -1 w β = W β D -1 w β = Π ′ so that
← -Π = Π : detailed balance also holds.

  to a scaling constant, to the weight W β (i n ) of the n-path i n . The invariant probability distribution µ * on {1, .., N } satisfying µ ′ * Π * = µ ′ * exists. It is given explicitly by µ * = π β * φ β and so: (25) µ * (i) = π β (i) φ β (i) , i = 1, .., N.

  j) log Π (i, j)β i µ (i) j Π (i, j) (U (j) -U (i)) =i µ (i) j Π (i, j) log Π (i, j) = s.


Alina Danciu

Quentin Gallis

Lucie Marie

Amélie Vairelles

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