Vlad Stefan Barbu

Slim Beltaief
email: beltaiefslim@hotmail.fr

Vlad Stefan Barbu
email: vladstefan.barbu@univ-rouen.fr

Serguei M Pergamenchtchikov
email: serge.pergamenshchikov@univ-rouen.fr

Adaptive efficient

come

In this paper we study the following linear regression model in continuous time

dy t =   q j=1 β j u j (t)   dt + dξ t , 0 ≤ t ≤ T , (1)
where the functions (u j ) 1≤j≤q are known linear independent 1-periodic R → R functions, the duration of observations T is an integer number and (ξ t ) t≥0 is an unobservable noise process defined in Section 2. The process (1) is observed only at the fixed time moments (y t j ) 0≤j≤n , t j = j p and n = pT ,

where the observations frequency p is some fixed integer number. We consider the model (1) in the case when the parameter dimension is grater than the number of observations, i.e., q > n. Such models are called big data or high dimension regression in continuous time (see, for example, in [START_REF] Fujimori | The Danzing selector for a linear model of diffusion processes[END_REF] for diffusion processes). The problem is to estimate the unknown parameters (β j ) 1≤j≤q on the basis of the observations (2). Usually for such problems one uses either the Lasso algorithm or the Dantzig selector method. It should be emphasized that to apply these methods one needs to assume sparsity conditions which provide the non large ("reasonable") number of the nonzero unknown parameters and, moreover, the parameter dimension q must be known (see, for example, in [START_REF] Hasttie | The Elements of Statistical Leaning. Data Mining, Inference and Prediction[END_REF]). It should be noted also that the case of unknown parameter dimension q is one of the crucial points in important practical problems such as, for example, signal and image statistical processing (see, for example, [START_REF] Beltaief | Model selection for the robust efficient signal processing observed with small Levy noise[END_REF] and the references therein). In this paper we study the model (1) in the nonparametric setting as the estimation problem for the function

S(t) = q j=1
β j u j (t) ,

i.e. dy t = S(t)dt + dξ t , 0 ≤ t ≤ T ,

where S is an unknown 1-periodic R → R function from L 2 [0, 1]. Here, we assume neither sparsity conditions, nor the condition that the parameter dimension is known, i.e., in particular, we can assume that q = +∞. Now the problem is to estimate the unknown function S in the model (3) on the basis of observations (2). Originally, such problems were considered in the framework "signal+white noise" models (see, for example, [START_REF] Ibragimov | Statistical Estimation: Asymptotic Theory[END_REF]; [START_REF] Kutoyants | Identification of dynamical systems with small noise[END_REF]; [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian white noise[END_REF]). Later, it was extended to the "color noise" models defined through non Gaussian Ornstein-Uhlenbeck processes [START_REF] Barndorff-Nielsen | Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial mathematics[END_REF]; Konev andPergamenshchikov (2012, 2015). The problem here is that the dependence defined on the basis of the Ornstein-Uhlenbeck processes disappears very fast, at a geometric rate. This means that such models are asymptotically equivalent to models with independent observations. To keep the dependence in the observations for large time periods for the estimation problem on the complete data in the paper Barbu, Beltaief and Pergamenshchikov (2019a) it is proposed to define the model (3) through semi-Markov processes with jumps. Such models considerably extend the potential applications of statistical results in many important practical fields such as finance, insurance, signals and image processing, reliability, biology (see, for example, Barbu, Beltaief and Pergamenshchikov (2019a); [START_REF] Barbu | Semi-Markov Chains and Hidden Semi-Markov Models toward Applications -Their use in Reliability and DNA Analysis[END_REF] and the references therein). In this paper we extend the semi-Markov regression models to the generalized semi-Markov processes by introducing an additional dependence in jump sizes of (ξ t ) t≥0 .

Methods

In this paper, in order to estimate the function S, we develop model selection methods using the quadratic risks defined as

R Q ( S T , S) = E Q,S S T -S 2 , f 2 = 1 0 f 2 (s)ds , (4)
where S T (•) is some estimate (i.e. any periodical function measurable with respect to the observations σ{y t 0 , . . . y t n }) and E Q,S is the expectation with respect to the distribution P Q,S of the process (3) corresponding to the unknown noise distribution Q in the Skorokhod space D[0, T ] and to the function S. We assume that this distribution belongs to some distribution family Q T specified in Section 2. To study the properties of the estimators uniformly over the noise distribution (what is really needed in practice), we use the robust risk defined as

R * T ( S T , S) = sup Q∈Q T R Q ( S T , S) . (5)
It should be noted that statistical procedures which are optimal in the sense of this risk possess stable mean square accuracy uniformly over all possible admissible noise distributions in the model (3). This means that the corresponding statistical optimal algorithms have high noise immunity and, therefore, significantly improve the quality and reliability of statistical inferences obtained on their basis.

To construct model selection procedures on the basis of the discrete data (2) we use the approach proposed in [START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF]. It should be noted that the main analytic tool in this paper is based on the exponential decrease rate of the dependence in Ornstein-Uhlenbeck models, and, therefore, we cannot apply these methods to semi-Markov models, which can retain a dependence in noises for a long time. So, in this paper, to study the estimation problem on the discrete observations (2) for the model (3) with noises defined through semi-Markov processes we develop new methods based on the special renewal theory from Barbu, Beltaief and Pergamenshchikov (2019a); based on these techniques we can analyse the approximation errors in the discrete observations and obtain non asymptotic sharp oracle inequalities. Moreover, as a consequence, we found constructive sufficient conditions on the observations frequency which provide the robust efficiency for proposed model selection procedures in adaptive setting, i.e. in the case when the regularity properties of the function S are unknown.

Main contributions of this paper

In this paper we use for the first time nonparametric adaptive methods for estimation problems in the framework of the big data generalized semi-Markov regression models. To this end we develop model selection procedures and corresponding analytical tools providing, under some constructive sufficient conditions, the optimality in the sharp oracle inequality sense and the robust adaptive efficiency in the minimax sense for the proposed estimators. It turns out that these conditions hold true for important practical cases such as, for example, regression models constructed through truncated fractional Poisson processes introduced in Barbu, Beltaief and Pergamenshchikov (2019b). Moreover, in this paper, we extend for the first time the model from Barbu, Beltaief and Pergamenshchikov (2019a) using the generalized semi-Markov models obtained by introducing a dependence structure in the sizes of the jumps. As an example, we use spherically symmetric random variables, which play very important role in many practical applications (see, for example, [START_REF] Fourdrinier | Improved selection model method for the regression with dependent noise[END_REF] and the references therein).

Organization of the paper

The rest of the paper is organized as follows. In Section 2 we state the main conditions under which we consider the model (3). In Section 3 we represent fractional Poisson processes and its main properties. In Section 4 we construct model selection procedures on the basis of weighted least squares estimates. In Section 5 we state the main results. In section 6 we develop the stochastic calculus for the generalized semi-Markov processes. Section 7 gives the proofs of the main results. Some auxiliary tools are given in an Appendix.

Main conditions

First, we assume that the noise process (ξ t ) t≥ 0 in the model ( 3) is defined as

ξ t = 1 w t + 2 L t + 3 z t , (6)
where 1 , 2 and 3 are unknown coefficients, (w t ) t≥ 0 is a standard Brownian motion,

L t = t 0 R *
x(µ(ds, dx) -µ(ds, dx)), µ(ds dx) is the jump measure with deterministic compensator µ(ds dx) = dsΠ(dx), Π(•) is the Lévy measure on R * = R\{0} (see, for example [START_REF] Liptser | Theory of Martingales[END_REF] for details), with Π(x 2 ) = 1 and Π(x 8 ) < ∞ .

(7)

Here we use the usual notations for Π(|x| m ) = R |z| m Π(dz). Note that Π(|x|) may be equal to +∞. In this paper we assume that the "dependent part" in the noise ( 6) is modelled by the generalized semi-Markov process (z t ) t≥ 0 defined as

z t = N t i=1 ζ i , (8)
where (ζ i ) i≥ 1 are random variables satisfying the following conditions:

C 1 ) ∀i ≥ 1 the expectations E ζ i = 0, E ζ 2 i = 1 and sup l≥1 E ζ 4 l < ∞; C 2 ) E ζ i ζ j = 0 for any i = j; C 3 ) For any 1 ≤ k 1 < k 2 < k 3 < k 4 the random variables (ζ k i ) 1≤i≤4 are such that E ζ ι 1 k 1 ζ ι 2 k 2 ζ ι 3 k 3 ζ ι 4 k 4
= 0 for any ι 1 , . . . , ι 4 ∈ {0, 1, 2, 3} for which 3 ≤ 4 i=1 ι i ≤ 4 and at least one among them is equal to one. Now we give some examples for the correlation conditions C 1 ) -C 3 ). To this end, we first remind the definition of spherically symmetric distribution (see, for example, in [START_REF] Fourdrinier | Improved selection model method for the regression with dependent noise[END_REF]

g(t) = E 1 (2πs) d/2 e -t 2s 2 , ( 9
)
where s is a non negative random variable. It should be emphasized that in radio-physics such distributions are very popular for statistical signal processing (see, for example, [START_REF] Middleton | Canonical non-Gaussian noise models: their implications for measurement and for prediction of receiver performance[END_REF]; [START_REF] Kassam | Signal detection in non-Gaussian noise[END_REF]). Using these definitions it is easy to see that the following random variables satisfy the conditions C 1 ) -C 3 ):

- In (8) the process N t is a general counting process defined as

N t = ∞ k=1 1 { k l=1 τ l ≤t} (10)
with (τ l ) l≥ 1 an i.i.d. sequence of positive integrated random variables with the distribution η and mean τ = E Q τ 1 > 0. We assume that the processes (N t ) t≥0 , (Y i ) i≥ 1 and (L t ) t≥0 are independent. In the sequel we will use the renewal measure defined as

η = ∞ l=1 η (l) , (11)
where η (l) is the lth convolution power of the measure η.

Remark 1 Note that in the case when the random variables (ζ j ) j≥1 are i.i.d. random variables, then (8) is the semi-Markov process used in Barbu, Beltaief and Pergamenshchikov (2019a).

To use the renewal methods from Barbu, Beltaief and Pergamenshchikov (2019a) we assume that the distribution η has a density g for which the following conditions hold true.

H 1 ) Assume that, for any x ∈ R, there exist the finite limits g(x-) = lim z→x-g(z) and g(x+) = lim z→x+ g(z) and, for any ∀K > 0, ∃δ = δ(K) > 0 for which

sup |x|≤K δ 0 |g(x + t) + g(x -t) -g(x+) -g(x-)| t dt < ∞. H 2 ) ∀γ > 0 the upper bound sup z≥0 z γ |2g(z) -g(z-) -g(z+)| < ∞. H 3 ) There exists β > 0 such that R + e βx g(x) dx < ∞.
H 4 ) ∃t * > 0 such that the Fourier transformation g(θ-it) belongs to L 1 (R) for any 0 ≤ t ≤ t * , where g(z) = (2π) -1 R e izv g(v)dv. Moreover, to check these conditions we will use the following assumption.

H * 4 ) The density g is two time continuously differentiable on R + with g(0) = 0 and there exists β > 0 such that +∞ 0

e βx g(x) + |g (x)| + |g (x)| dx < ∞ and lim x→∞ e βx g(x) + |g (x)| = 0.
It is clear that the conditions H 1 )-H 3 ) hold true in this case. To obtain the condition H 4 ) it suffices to calculate the integral in g, integrating by parts two times. For example, one can take gamma distribution of order m ≥ 2

g(x) = a m x m-1 m! e -ax 1 {x≥0} and a > 0 . ( 12
)
It should be noted that in view of Proposition 5.2 from Barbu, Beltaief and Pergamenshchikov (2019a), Conditions H 1 )-H 4 ) imply that the renewal measure (11) has a continuous density ρ such that

Υ 1 = +∞ 0 |Υ (x)| dx < ∞ and Υ (x) = ρ(x) - 1 τ . ( 13
)
Remark 2 It should be noted that Condition H 4 ) does not hold for the exponential random variable (τ j ) j≥1 since its density is not continuous in zero. But for exponential random variables, i.e. in the case when (N t ) t≥0 is a Poisson process, the renewal density can be calculated directly, i.e. ρ(x) ≡ 1/τ and Υ ≡ 0.

Now we describe the class of possible admissible noise distributions used in the robust risk (5). To this end we set

σ Q = 2 1 + 2 2 + 2 3 τ . ( 14
)
As to the parameters in (6), we assume that

ς * ≤ σ Q ≤ ς * , (15)
where the unknown bounds 0 < ς * ≤ ς * can be functions of T , i.e. ς * = ς * (T ) and ς * = ς * (T ), such that for any b > 0 lim

T →∞ T b ς * (T ) = +∞ and lim T →∞ ς * (T ) T b = 0 . ( 16
)
We denote by Q T the family of all distributions of the process (6) in D[0, T ] satisfying the properties ( 15) -( 16).

Remark 3 As we will see later, the parameter ( 14) is the limit of the Fourier transform of the noise process ( 6). This limit is called variance proxy (see [START_REF] Konev | Efficient robust nonparametric in a semimartingale regression model[END_REF]).

Truncated fractional Poisson processes

As an example of the process (10) satisfies the conditions H 1 ) -H 4 ) we give the truncated fractional Poisson process introduced in Barbu, Beltaief and Pergamenshchikov (2019b). To this end, we remind the definition of the fractional Poisson process (see, for example, Biard and Saussereaur (2014); [START_REF] Laskin | Fractional Poisson processes[END_REF]). The process ( 10) is called fractional Poisson process if the i.i.d. random variables (τ j ) have the Mittag-Leffler distribution which, for some a > 0, is defined as

P(τ 1 > t) = E H (-at H ) , (17)
where 0 < H ≤ 1 is called the Hurst index,

E H (z) = ∞ k=0 z k Γ (1 + Hk) and Γ (x) = +∞ 0 t x-1 e -t dt .
Note that, if H = 1, then we obtain the exponential distribution with parameter a > 0 and, therefore, the process ( 10) is a Poisson process. If 0 < H < 1, then the density of the distribution (17) (see, for example, [START_REF] Repin | Fractional Poisson law. -Radiophys[END_REF]) can be represented as

f H (t) = a sin(πH) π +∞ 0 x H e -tx x 2H + a 2 + 2ax H cos(πH) dx . (18)
Form here we can directly obtain that

f H (t) ∼ t H-1 , f H (t) ∼ t H-2 , f H (t) ∼ t H-3 as t → 0 (19)
and

f H (t) ∼ t -H-1 , f H (t) ∼ t -H-2 , f H (t) ∼ t -H-3 as t → ∞ . (20)
In particular, this implies that the Mittag-Leffler distribution has a heavy tail, i.e.

P(τ 1 > t) ∼ t -H as t → ∞ , (21)
i.e. Eτ 1 = +∞. Therefore, the condition H 3 ) does not hold for the distribution (17). To correct this effect, in Barbu, Beltaief and Pergamenshchikov (2019b) it is proposed to replace the Mittag-Leffler random variables in (10) with i.i.d. random variables distributed as

τ * 1 = min(X b * , X * )
, where X * is a Mittag-Leffler with 0 < H < 1, 0 < b ≤ H/3 and X * is a positive random variable satisfying the condition H * 4 ). Such processes are called truncated Poisson processes. Using the asymptotic properties ( 19) and ( 20) one can check directly that the random variable τ * 1 satisfies the condition H * 4 ) and, therefore, the conditions H 1 ) -H 4 ) hold true for this case.

Remark 4 It should be noted also that the process (10) with the Mittag-Leffler random variables has a "memory" in its increments (see, for example, Maheshwari and Vellaisamy ( 2016)) in the sense that, for any δ > 0 and s > 0, the correlation coefficient

Corr N s+δ -N s , N t+δ -N t ∼ t -3-H 2 as t → ∞ .
It should be noted that this property is very important for many practical problems and allows essentially to expand the possible applications of statistical results. Unfortunately, we can't use directly the fractional Poisson process in the regression model (3) since the impulse noise of the fractional Poisson processes will be very rare, since the time between jumps is not integrable, i.e. very large and, therefore, they have almost negligible influence in the observation models. On the contrary, the truncated process has an exponential moment, i.e. the same property as Poisson processes, and, moreover, it keeps a dependence on large time intervals.

Model selection

In this section we construct a model selection procedure for estimating the unknown function S given in (3) starting from the discrete-time observations

(2) and we establish the oracle inequality for the associated risk. To this end, note that for any function f

: [0, T ] → R from L 2 [0, T ], the integral I T (f ) = T 0 f (s)dξ s (22)
is well defined, with E Q I T (f ) = 0. Moreover, as it is shown in Lemma 1 under the conditions H 1 )-H 4 ),

E Q I 2 T (f ) ≤ κ Q T 0 f 2 s d s and κ Q = 2 1 + 2 2 + 2 3 |ρ| * ( 23
)
where

|ρ| * = sup t≥0 |ρ(t)| < ∞.
In this paper we will use the trigonometric basis (φ j ) j≥ 1 in L 2 [0, 1] defined as

φ 1 = 1 , φ j (x) = √ 2Tr j (2π[j/2]x) , j ≥ 2 , ( 24
)
where the function Tr j (x) = cos(x) for even j and Tr j (x) = sin(x) for odd j, [x] denotes the integer part of x. Note, that these functions are orthonormal on the points (t j ) 1≤j≤p , i.e. for any 1 ≤ i, j ≤ p

(φ i , φ j ) p = 1 p p l=1 φ i (t l )φ j (t l ) = 1 {i=j} . ( 25
)
In the sequel we denote by x 2 p = (x, x) p . Now note that, for any 1 ≤ l ≤ p, S(t l ) = p j=1 θ j,p φ j (t l ) and θ j,p = (S, φ j ) p .

(26)

Using the approach from [START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF], we estimate the Fourier coefficients θ j,p as

θ j,p = 1 T T 0 ψ j,p (t)d y t , and ψ j,p (t) = n l=1 φ j (t l )1 {t l-1 <t≤t l } . ( 27
)
It is clear that the functions (ψ j,p

) 1≤j≤p are orthonormal in L 2 [0, 1], i.e. (ψ j,p , ψ i,p ) = 1 0 ψ j,p (t)ψ i,p (t)d t = (φ j , φ i ) p = 1 {i=j} . ( 28
)
The Fourier coefficients of S in the basis can be represented as

θ j,p = (S, ψ i,p ) = 1 0 S(t)ψ i,p (t)d t = θ j,p + h j,p , (29)
where h j,p (S) =

p l=1 t l t l-1 φ j (t l )(S(t) -S(t l ))d t. Therefore, (27) implies θ j,p = θ j,p + 1 √ T ξ j,p and ξ j,p = 1 √ T I T (ψ j,p ) . (30)
As in Barbu, Beltaief and Pergamenshchikov (2019a) we use the model selection procedures based on the following weighted least squares estimators

S λ (t) = p j=1 λ(j) θ j,p ψ j,p (t) , 0 ≤ t ≤ 1 , (31)
where the weight vector λ = (λ(1), . . . , λ(p)) belongs to some finite set Λ from [0, 1] p . Here the prime denotes the transposition. Moreover, we set

m * = card(Λ) and Λ * = max λ∈Λ p j=1 1 {λ(j)>0} , (32)
where card(Λ) is the cardinal number of the set Λ. We assume that Λ * ≤ n. Now we use the same criteria as in Barbu, Beltaief and Pergamenshchikov (2019a) to chose a weight vector in Λ, i.e.we minimize the empirical error

Err(λ) = S λ -S 2 , ( 33
)
which can be represented as

Err(λ) = p j=1 λ 2 (j) θ 2 j,p -2 p j=1 λ(j) θ j,p θ j,p + S 2 . ( 34
)
Note that the Fourier coefficients (θ j ) j≥ 1 are unknown. Therefore, using the approach from Barbu, Beltaief and Pergamenshchikov (2019a) to minimize this function we replace the terms θ j,p θ j,p by their estimators

θ j,p = θ 2 j,p - σ Q T ,
where the proxy variance σ Q is defined in (15). In the case when this variance is unknown we use its estimator, i.e.

θ j,p = θ 2 j,p - σ T T and σ T = T p p j=[ √ T ] θ 2 j,p . (35)
Now, using this estimator we define the penalty term as

P T (λ) = σ T |λ| 2 T and |λ| 2 = p j=1 λ 2 (j) . ( 36
)
In the case, when the variance σ Q is known we set

P T (λ) = σ Q |λ| 2 T . (37)
Finally, we define the cost function as

J T (λ) = p j=1 λ 2 (j) θ 2 j,T -2 p j=1 λ(j) θ j,T + δ P T (λ), (38)
where δ > 0 is some threshold which will be specified later. Now we set the model selection procedure as

S * = S λ and λ = argmin λ∈Λ J T (λ) . (39)
In the case when λ is not unique we take one of them.

5 Main results

Oracle inequalities

Firstly, we obtain the non asymptotic oracle inequality for the model selection procedure (39). To this end we need a condition for the observations frequency.

H 5 ) Assume that the frequency p is a function of T , i.e. p = p T , such that lim inf

T →∞ p T T 5/6 > 0 and lim sup

T →∞ p T T < ∞ . ( 40
)
Theorem 1 Assume that the conditions C 1 ) -C 3 ) and H 1 )-H 5 ) hold true.

Then, there exists some constant c * > 0 such that for any T ≥ 1 and any noise distribution Q ∈ Q T and 0 < δ ≤ 1/6, the procedure (39) satisfies the following oracle inequality

R Q ( S * , S) ≤ 1 + 3δ 1 -3δ min λ∈Λ R Q ( S λ , S) + c * m * δT 1 + σ 2 Q + Λ * E Q | σ T -σ Q | . (41)
In the case when σ Q is known the inequality (41) has the following form.

Corollary 1 Assume that the conditions C 1 ) -C 3 ) and H 1 )-H 5 ) hold true and that the proxy variance σ Q is known. Then there exists some constant c * > 0 such that for any T ≥ 1 and for any noise distribution Q ∈ Q T and 0 < δ ≤ 1/6, the procedure (39) with σ T = σ Q , satisfies the following oracle inequality

R Q ( S * , S) ≤ 1 + 3δ 1 -3δ min λ∈Λ R Q ( S λ , S) + c * (1 + σ 2 Q )m * δT .
Now we study the estimator σ T defined in (35).

Proposition 1 Assume that the conditions C 1 ) -C 3 ) and H 1 ) -H 5 ) hold true for the model (3) and that S(•) is continuously differentiable. Then, there exists a constant c * > 0 such that for any

T ≥ 2, Q ∈ Q T and p > √ T , E Q,S | σ T -σ Q | ≤ c * (1 + | Ṡ| 2 )(1 + σ Q ) 2 g * T,p , (42)
where

g * T,p = √ T /p + 1/ √ p.
Now Theorem 1 and this proposition imply directly the following result.

Theorem 2 Assume that the function S is continuously differentiable and that the conditions C 1 ) -C 3 ) and H 1 )-H 5 ) hold true. Then there exists some constant c * > 0 such that for any continuously differentiable function S for any T ≥ 2, for any noise distribution

Q ∈ Q T , p > √ T and 0 < δ ≤ 1/6, R Q ( S * , S) ≤ 1 + 3δ 1 -3δ min λ∈Λ R Q ( S λ , S) + c * m * δT (1 + σ Q ) 2 1 + | Ṡ| 2 1 + Λ * g * T,p .
To study robust properties of the procedure (39) we need a condition for weights. Theorem 3 Assume that the function S is continuously differentiable, the conditions the conditions C 1 ) -C 3 ) and H 1 )-H 6 ) hold true. Then, the procedure (39) for any T ≥ 2, p > √ T and 0 < δ < 1/6 satisfies the following oracle inequality

R * ( S * , S) ≤ 1 + 3δ 1 -3δ min λ∈Λ R * ( S λ , S) + U * T T δ ,
where the term U * T > 0 is such that for any r > 0 and b > 0, lim

T →∞ sup Ṡ ≤r T -b U * T = 0 . ( 43
)
In order to obtain the efficiency property, we specify the weight coefficients in the procedure (39). Consider, for some fixed 0 < ε < 1, a numerical grid of the form

A = {1, . . . , k * } × {ε, . . . , mε} , m = [1/ε 2 ] , (44)
where k * ≥ 1 and ε are functions of T , i.e. k * = k * (T ) and

ε = ε(T ), such that      lim T →∞ k * (T ) = +∞ , lim T →∞ k * (T ) ln T = 0 , lim T →∞ ε(T ) = 0 and lim T →∞ T b ε(T ) = +∞ (45)
for any b > 0. One can take, for example, for T ≥ 2

ε(T ) = 1 ln T and k * (T ) = k * 0 + √ ln T ,
where k * 0 ≥ 0 is a fixed constant. For each α = (k, r) ∈ A, we set the vector

λ α = (λ α (j)) 1≤j≤p
through its components which are defined as

λ α (j) = 1 {1≤j<ln T } + 1 -(j/ω α ) k 1 {ln T ≤j≤ω α } ,
where

ω α = (k + 1)(2k + 1) π 2k k rυ T 1/(2k+1)
, υ T = T /ς * and ς * is introduced in (15). Now we define the set Λ as

Λ = {λ α , α ∈ A} . (46)
These weight coefficients are used in Konev Pergamenshchikov (2009a); Konev andPergamenshchikov (2012, 2015) for continuous time regression models to show the asymptotic efficiency. Note also that in this case the cardinal of the set Λ is m * = k * m. Moreover, taking into account that for k ≥ 1 the coefficient ω α < (rυ T ) 1/(2k+1) , we obtain that the norm of the set Λ defined in (32) can be bounded as Λ * ≤ sup α∈A ω α ≤ (υ T /ε) 1/3 . Therefore, the properties (45) imply the condition H 6 ).

Robust asymptotic efficiency

Now we study the asymptotic efficiency properties for the procedure (39), ( 46) with respect to the robust risks (5) defined by the distribution family ( 15) -( 16). To this end, we assume that the unknown function S in the model (3) belongs to the Sobolev ball

W r,k =    f ∈ C k per [0, 1] : k j=0 f (j) 2 ≤ r    , (47)
where r > 0, k ≥ 1 are some unknown parameters,

C k per [0, 1] is the set of k times continuously differentiable functions f : [0, 1] → R such that f (i) (0) = f (i) (1) for all 0 ≤ i ≤ k. Note, that the class (47) is an ellipsoid, i.e. W r,k =    f = j≥1 θ j φ j : ∞ j=1 a j θ 2 j ≤ r    ( 48
)
where

a j = k i=0 (2π[j/2]) 2i .
Similarly to Barbu, Beltaief and Pergamenshchikov (2019a) we will show here that the asymptotic sharp lower bound for the normalized robust risk ( 5) is given by the well-known Pinsker constant defined as

l * = l * (r) = ((2k + 1)r) 1/(2k+1) k (k + 1)π 2k/(2k+1)
.

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To study efficient properties we need to use the set Ξ T of all possible estimators S T measurable with respect to the sigma-algebra σ{y t , 0 ≤ t ≤ T }.

Theorem 4 For the risk (5) with the coefficient rate υ T = T /ς * lim inf

T →∞ υ 2k/(2k+1) T inf S T ∈Ξ T sup S∈W r,k R * T ( S T , S) ≥ l * . ( 50
)
Note that, if the radius r and the regularity k are known, i.e. for the nonadaptive estimation problem on the continuous observations (y t ) 0≤t≤T , in Barbu, Beltaief and Pergamenshchikov (2019a) it is proposed to use the estimate S λ 0 defined in (31) with the weights ( 46)

λ 0 = λ α 0 , α 0 = (k, r 0 ) and r 0 = [r/ε]ε . (51)
Now, we show the same result for the discrete observations (2).

Proposition 2 Assume that the conditions the conditions C 1 ) -C 2 ) and H 1 )-H 5 ) hold true. Then

lim T →∞ υ 2k/(2k+1) T sup S∈W r,k R * T ( S λ 0 , S) ≤ l * . ( 52
)
For the adaptive estimation we user the model selection procedure (39) with the parameter δ defined as a function of T , i.e. δ = δ T , such that lim

T -→∞ δ T = 0 and lim T -→∞ T b δ T = +∞ (53)
for any b > 0. For example, we can take δ T = (6 + ln T ) -1 .

Theorem 5 Assume that the conditions C 1 ) -C 3 ) and H 1 )-H 6 ) hold true.

Then the robust risk (5) for the procedure (39) with the coefficients (46) and the parameter δ = δ T satisfying (53) has the following upper bound lim sup

T →∞ υ 2k/(2k+1) T sup S∈W r,k R * T ( S * , S) ≤ l * .
Theorem 4 and Theorem 5 imply the following result.

Theorem 6 Assume that the conditions C 1 ) -C 3 ) and H 1 )-H 6 ) hold true.

Then the procedure (39) with the weight coefficients (46) and the parameter δ = δ T satisfying (53) is asymptotically efficient, i.e.

lim T →∞ inf S T ∈Ξ T sup S∈W r,k R * T ( S T , S) sup S∈W r,k R * T ( S * , S) = 1
and lim

T →∞ υ 2k/(2k+1) T sup S∈W r,k R * T ( S * , S) = l * .
Remark 5 It is well known that the optimal (minimax) risk convergence rate for the Sobolev ball W r,k is T 2k/(2k+1) (see, for example, [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian white noise[END_REF]; Konev Pergamenshchikov (2009b)). We see here that the efficient robust rate is υ 2k/(2k+1) T

, i.e. if the distribution upper bound ς * → 0 as T → ∞ we obtain a faster rate with respect to T 2k/(2k+1) , and if ς * → ∞ as T → ∞ we obtain a slower rate. In the case when ς * is constant the robust rate is the same as the classical non robust convergence rate.

Big data analysis for the model (1)

Now we consider the estimation problem for the parameters (β j ) 1≤j≤q in (3) with unknown q. In this case we have to estimate the sequence β = (β j ) j≥1 in which β j = 0 for j ≥ q + 1. To this end we assume that the functions (u j ) j≥1 are orthonormal in L 2 [0, 1], i.e. (u i , u j ) = 1 {i =j} . Indeed, we can use always the Gram-Schmidt orthogonalization procedure to provide this property. Thus, in this case we estimate the parameters β = (β j ) j≥1 through the estimator (39) as β * = ( β * ,j ) j≥1 and β * ,j = (u j , S * ). Similarly, using the weighted estimators (31) we define the basic estimators ( β λ ) λ∈Λ as β λ = ( β j,λ ) j≥1 and β j,λ = (u j , S λ ). Taking into account that in this case

| β * -β| 2 = ∞ j=1 ( β * ,j -β j ) 2 = S * -S 2 and | β λ -β| 2 = S λ -S 2 ,
Theorem 3 implies the following oracle inequality.

Theorem 7 Assume that the function (3) is continuously differentiable and the conditions C 1 ) -C 3 ), H 1 )-H 5 ) and ( 15)-( 16) hold true. Then, for any n ≥ 1 and 0 < δ < 1/6, the following oracle inequality holds true

sup Q∈Q T E Q,S | β * -β| 2 ≤ 1 + 3δ 1 -3δ min λ∈Λ sup Q∈Q T E Q,S | β λ -β| 2 + U * T T δ ,
where the term U * T > 0 satisfies the property (43).

Moreover, Theorem 6 implies the following efficiency property.

Theorem 8 Assume that the conditions C 1 ) -C 3 ) and H 1 )-H 6 ) hold true.

Then the estimator β * constructed through the procedure (39) with the weight coefficients (46) and the parameter δ = δ T satisfying (53) is asymptotically efficient in the minimax sense, i.e.

lim T →∞ inf β T sup S∈W r,k sup Q∈Q T E Q,S | β T -β| 2 sup S∈W r,k sup Q∈Q T E Q,S | β * -β| 2 = 1 (54)
and lim

T →∞ υ 2k/(2k+1) T sup S∈W r,k sup Q∈Q T E Q,S | β * -β| 2 = l * ,
where the infimum is taken over all possible estimators β T measurable with respect the field σ{y t , 0 ≤ t ≤ T } and the lower bound l * is defined in (49).

Remark 6 It should be emphasized that the efficiency properties ( 54) are obtained without sparse conditions on the number of non zero parameters β j in the model (1) (see, for example, in [START_REF] Hasttie | The Elements of Statistical Leaning. Data Mining, Inference and Prediction[END_REF]). Moreover, we do not use even the parameter dimension q which can be equal to +∞.

Stochastic calculus for generalized semi-Markov processes

In this section we study some properties of the stochastic integrals ( 22). First, note that using the conditions C 1 ) and C 2 ) and the stochastic calculus developed in Barbu, Beltaief and Pergamenshchikov (2019a) for semi-Markov processes we can show the following Lemmas 1 and 2.

Lemma 1 Assume that the conditions C 1 ) -C 2 ) and H 1 )-H 4 ) hold true.

Then, for any non random functions f and h from L 2 [0, T ]

E Q I t (f )I t (h) = ( 2 1 + 2 2 ) (f, h) t + 2 3 (f, hρ) t , (55)
where (f, h) t = t 0 f (s) h(s)ds and ρ is the density of the renewal measure (11).

It should be noted that this lemma implies directly that the stochastic integral ( 22) satisfies the properties (23).

Lemma 2 Assume that the conditions C 1 ) -C 2 ) and H 1 )-H 4 ) hold true. Then, for any bounded [0, ∞) → R functions f and h and for any k ≥ 1,

E Q I t k -(f ) I t k -(h) | G = ( 2 1 + 2 2 )(f , h) t k + 2 3 k-1 l=1 f (t l ) h(t l ),
where

t k = k j=1 τ j and G = σ{t l , l ≥ 1}.
Lemma 3 Assume that the conditions C 1 ) -C 3 ) and H 1 )-H 4 ) hold true.

Then, for any nonrandom bounded [0, T ] → R functions f and h, the expectation

E Q T 0 I 2 t-(f )I t-(h)h(t)dξ t = 0.
Proof. Setting Ľt = 1 w t + 2 L t , we can represent the integral ( 22) as

I t (f ) = Ǐt (f ) + 3 I z t (f ) , (56)
where Ǐt (f ) = t 0 f (u)d Ľu and

I z t (f ) = t 0 f (u)dz u .
Note here, that using the condition ( 7) and the inequality for martingales from [START_REF] Novikov | On discontinuous martingales[END_REF] we can obtain that

E Q sup 0≤t≤T Ǐ8 t (f ) < ∞. Since Ľt and z t are independent, we get E Q T 0 I 2 t-(f )I t-(h)h(t)d Ľt = 0. Moreover, the conditions C 1 ) -C 3 ) yield, that for any non random (c i,j ) and k ≥ 1 E k-1 j=1 c 1,j ζ j 2 ζ k = 0 and E k-1 j=1 c 1,j ζ j 2 k-1 j=1 c 2,j ζ j ζ k = 0.
Therefore, taking into account that the sequence (ζ k ) k≥1 does not depend on the moments (t k ) k≥1 and the process ( Ľt ) t≥0 , and using the same method as in the proof of Lemma 8.4 from Barbu, Beltaief and Pergamenshchikov (2019a) we obtain

E Q T 0 I 2 t-(f )I t-(h)h(t)dz t = E Q k≥1 1 {t k ≤T } I 2 t k -(f )I t k -(h)h(t k )ζ k = 0 .
This implies Lemma 3. Now we study the integrals defined in (64) as functions of f .

Proposition 3 Assume that the conditions C 1 ) -C 3 ) and H 1 )-H 4 ) hold true. Then, for any [0, ∞) → R functions f, h such that |f | * ≤ 1 and |h| * ≤ 1, one has |E Q I T (f ) I T (h)| ≤ 12σ 2 Q (1 + τ ) 2 (f, h) 2 T + T c , (57)
where c = (2 + Π(x 4 ) + 2|ρ| * )(1 + Υ 2 1 ) and |f | * = sup t≥0 |f (t)|.
Proof. First of all, note that in view of the Ito formula and using the fact that for the process (6) the jumps ∆z s ∆L s = 0 a.s. for any s ≥ 0, we obtain that

dI 2 t (f ) = 2I t-(f )dI t (f ) + 2 1 f 2 (t)d t + 2 2 d 0≤s≤t f 2 (s)(∆L s ) 2 + 2 3 d 0≤s≤t f 2 (s)(∆z s ) 2 .
Note also that Lemma 1 yields

E Q I 2 t (f ) = ( 2 1 + 2 2 ) f 2 t + 2 3 f √ ρ 2 t with f 2 t = t 0 f 2 (t)dt. Therefore, d I t (f ) = 2I t-(f )f (t)dξ t + f 2 (t)d m t , m t = 2 2 mt + 2 3 m t ,
where mt = 0≤s≤t (∆L s ) 2 -t and m t = 0≤s≤t (∆z s ) 2 -t 0 ρ(s)ds. Thus,

E Q I T (f ) I T (h) = E Q T 0 I t-(f )d I t (h)+E Q T 0 I t-(h)d I t (f )+E Q [ I(f ), I(h) ] T .
Using here Lemma 3 and, taking into account that ( mt ) t≥0 is a square integrated martingale, we get

E Q T 0 I t-(f )d I t (h) = E Q T 0 I t-(f )h 2 (t)d m t = ρ 2 3 E Q T 0 I 2 t-(f )h 2 (t)dm t .
The last integral can be represented as

E Q T 0 I 2 t-(f )h 2 (t)dm t = J 1 -J 2 , (58)
where

J 1 = E Q k≥1 I 2 t k -(f )h 2 (t k )1 {t k ≤T } and J 2 = T 0 E Q I 2 t (f )h 2 (t)ρ(t)dt. By Lemma 2 we get J 1 = E Q k≥1 E Q I 2 t k -(f )|G h 2 (t k )1 {t k ≤T } = ( 2 1 + 2 2 )J 1,1 + 2 3 J 1,2 ,
where

J 1,1 = E Q k≥1 f 2 t k h 2 (t k )1 {t k ≤T } = T 0 f 2 t h 2 (t)ρ(t)dt and J 1,2 = E Q k≥1 k-1 l=1 f 2 (t l ) h 2 (t k )1 {t k ≤T } = E Q l≥1 f 2 (t l ) k≥l+1 h 2 (t k )1 {t k ≤T } = T 0 f 2 (x) T -x 0 h 2 (x + t)ρ(t)dt ρ(x)dx .
Moreover, using Lemma 1 for the last term in (58), we obtain that

J 2 = ( 2 1 + 2 2 ) T 0 f 2 t h 2 (t)ρ(t)dt + 2 3 T 0 f √ ρ 2 t h 2 (t)ρ(t)dt
and we can represent the expectation in ( 58) as

E Q T 0 I 2 t-(f )h 2 (t)dm t = 2 3 T 0 f 2 (x) T x h 2 (t)(Υ (t -x) -Υ (t))dt ρ(x)dx ,
i.e. we can estimate this as

|E Q T 0 I 2 t-(f )h 2 (t)dm t | ≤ 2 2 3 T Υ 1 . Therefore, |E Q T 0 I t-(f )d I t (h)| + |E Q T 0 I t-(h)d I t (f )| ≤ 4 4 3 T Υ 1 . (59)
Furthermore, note that

[ I(f ), I(h)] T =< I c (f ), I c (h) > T +D T (f, h) ,
where

I c t (f ) = 2 1 t 0 I s (f )f (s)dw s and D T (f, h) = 0≤t≤T ∆ I d t (f )∆ I d t (h). In this case I d t (f ) = 2 t 0 I s-(f )f (s)dξ d s + t 0 f 2 (s)d m s and ξ d t = 2 L t + 3 z t . Therefore, in view of Lemma 1, E Q < I c (f ), I c (h) > T = 4ρ 2 1 T 0 E Q (I t (f )I t (h))f (t)h(t)dt = 4ρ 2 1 (ρ 2 1 + ρ 2 2 ) T 0 (f, h) t f (t)h(t)dt + 4ρ 2 1 ρ 2 3 T 0 (f, hρ) t f (t)h(t)dt = 4ρ 2 1 σ Q (f, h) 2 T + 4ρ 2 1 ρ 2 3 T 0 (f, gΥ ) t f (t)h(t)dt . Since |f | * ≤ 1 and |h| * ≤ 1, we get T 0 |(f, hΥ ) t f (t)h(t)|dt ≤ T Υ 1 and E Q < I c (f ), I c (h) > T ≤ 4σ 2 Q (f, h) 2 T + T τ Υ 1 . (60)
To study the process D T (f, h) note that ∆ξ d t ∆ m t = 3 2 (∆L t ) 3 + 3 3 (∆z t ) 3 . Note also that for any t ≥ 0 the expectation E Q I t (f ) = 0. Therefore, using the definition of the process L t we obtain through the Fubini theorem, that for any bounded [0, T ] → R measurable non random functions V

E Q 0≤t≤T V (t) I t-(f )(∆L t ) 3 = Π(x 3 ) T 0 V (t)E Q I t (f )dt = 0 .
Moreover, since the processes ( Ľt ) t≥0 and (z t ) t≥0 are independent we get

E Q 0≤t≤T V (t) Ǐt-(f )(∆z t ) 3 = E Q k≥1 V (t k )ζ 3 k E Q Ǐt k -(f )|G z = 0 ,
where the integral Ǐt (f ) is defined in ( 56) and the G z = σ{z t , t ≥ 0}. Note that the condition C 3 ) implies that for any k ≥ 1 and nonrandom (c j ) j≥1

E Q k-1 j=1 c j ζ j ζ 3 k = 0. Therefore, E Q 0≤t≤T V (t)I z t-(f )(∆z t ) 3 = 0 and E Q 0≤t≤T I t-(f )f (t)h 2 (t)∆ξ d t ∆ m t = E Q 0≤t≤T I t-(h)h(t)f 2 (t)∆ξ d t ∆ m t = 0 .
So, the expectation of D T (f, h) can be represented as

E Q D T (f, h) = 4 2 2 E Q D 1,T (f, h) + 4 2 3 E Q D 2,T (f, h) + E Q D 3,T (f, h) , where D 1,T (f, h) = 0≤t≤T I t-(f )I t-(h)f (t)h(t)(∆L t ) 2 , D 2,T (f, h) = 0≤t≤T I t-(f )I t-(h)f (t)h(t)(∆z t ) 2 and D 3,T (f, h) = 0≤t≤T f 2 (t) h 2 (t)(∆ m t ) 2 . First, since Π(x 2 ) = 1, we get E Q D 1,T (f, h) = T 0 f (t)h(t)E Q I t (f )I t (h) dt = ( 2 1 + 2 2 ) T 0 f (t)h(t) (f, h) t dt + 2 3 T 0 f (t)h(t) (f, hρ) t dt = σ Q (f, h) 2 T + 2 3 T 0 f (t)h(t) (f, hΥ ) t dt and |E Q D 1,T (f, h)| ≤ σ Q (f, h) 2 T + T τ Υ 1 .
Then, taking into account that E ζ 2 j = 1 and using Lemma 2, we represent the expectation for D 2,T (f, h) as

E Q D 2,T (f, h) = E k≥1 E Q I t k -(f )I t k -(h)|G f (t k )h(t k ) 1 {t k ≤T } = ( 2 1 + 2 2 )E Q k≥1 (f , h) t k f (t k )h(t k ) 1 {t k ≤T } + 2 3 E Q D 2,T (f, h) = ( 2 1 + 2 2 ) T 0 (f, h) t f (t)h(t)ρ(t)dt + 2 3 E Q D 2,T (f, h) , where D 2,T (f, h) = k≥1 k-1 l=1 f (t l ) h(t l )f (t k )h(t k ) 1 {t k ≤T } . Note, that T 0 (f, h) t f (t)g(t)ρ(t)dt = 1 2τ (f, h) 2 T + T 0 (f, h) t f (t)g(t)Υ (t)dt , i.e. T 0 (f, h) t f (t)g(t)ρ(t)dt ≤ 1 2τ (f, h) 2 T + T Υ 1 .
Furthermore, the expectation of D 2,T (f, h) can be represented as

E Q D 2,T (f, h) = E Q l≥1 f (t l ) h(t l ) k≥l+1 f (t k )h(t k ) 1 {t k ≤T } = T 0 f (x)g(x) T -x 0 f (x + t)h(x + t)ρ(t)dt ρ(x)dx = 1 2τ 2 (f, h) 2 T + D 2,T (f, h) ,
where

D 2,T (f, h) = T 0 f (x)h(x) T -x 0 f (x + t)h(x + t)Υ (t)dt ρ(x)dx + 1 τ T 0 f (x)h(x) T -x 0 f (x + t)h(x + t)Υ (t)dt Υ (x)dx . This implies |D 2,T (f, h)| ≤ T (|ρ| * + 1/τ )(1 + Υ 2 1
) and, therefore,

|E Q D 2,T (f, h)| ≤ σ Q (f, h) 2 T τ + T (2 + τ |ρ| * )(1 + Υ 2 1 )
.

Moreover, we can calculate directly

E Q D 3,T (f, h) = 4 2 Π(x 4 ) T 0 f 2 (t) h 2 (t)dt + 4 3 T 0 f 2 (t) h 2 (t)ρ(t)dt and E Q D 3,T (f, h) ≤ T σ 2 Q Π(x 4 ) + |ρ| * (1 + τ ) 2
. From here we obtain that

|E Q D T (f, h)| ≤ σ 2 Q (1+τ ) 2 4(f, h) 2 T + T c
, where c is given in (57). From this and (60) we find

E Q [ I(f ), I(h)] T ≤ 8σ 2 Q (1 + τ ) 2 (f, h) 2 T + T c
. This bound and ( 59) implies (57). Hence the proof is achieved.

In order to prove the oracle inequalities we need to study the conditions introduced in [START_REF] Konev | Efficient robust nonparametric in a semimartingale regression model[END_REF] for the general semimartingale model (3). To this end, we set for any x ∈ R p the functions

B 1,Q (x) = p j=1 x j E Q ξ 2 j,p -σ Q and B 2,Q (x) = p j=1 x j ξ j,p , (61)
where σ Q is defined in (15) and ξ j,p = ξ 2 j,p -E Q ξ 2 j,p .

Proposition 4 Assume that the conditions C 1 )-C 2 ), H 1 )-H 5 ) hold true.

Then there exists some constant c * > 0 such that for any

Q ∈ ∪ k≥1 Q k L 1,Q = sup T ≥3 sup x∈[-1,1] p B 1,Q (x) < c * σ Q (62) and L 2,Q = sup T ≥3 sup |x|≤1 E Q B 2 2,Q (x) ≤ c * σ 2 Q , (63)
where

|x| 2 = p j=1 x 2 j .
Proof. Firstly, using here Lemma 1, we obtain that

Eξ 2 j,p = 2 1 + 2 2 + 2 3 T T 0 ψ 2 j,p (x) ρ(x)d x = σ Q + 2 3 T T 0 ψ 2 j,p (x) Υ (x)d x .
From (13) it follows that E Q ξ 2 j,p -σ Q ≤ 2 2 3 Υ 1 /T and, therefore, the condition H 5 ) implies the inequality (62). Next, note that

E Q   p j=2 x j ξ j,p   2 ≤ 1 T 2 p j=1 p l=1 |x j | |x l ||E Q I T (ψ j,p ) I T (ψ l,p )| , (64)
where

I T (f ) = I 2 T (f ) -E Q I 2 T (f )
. Now Proposition 3 and the property (28) imply, that for some constant c * > 0 and for |x| ≤ 1

E Q   p j=2 x j ξ j,p   2 ≤ c *   |x| 2 + 1 T   p j=1 |x j |   2    ≤ c * 1 + p T .
The condition H 5 ) implies directly Proposition 4.

Proofs

Proof of Theorem 1

Using the cost function given in (38), we can rewrite the empirical squared error in (34) as follows

Err(λ) = J T (λ) + 2 p j=1 λ(j)ϑ j,p + S 2 -δ P T (λ), (65)
where

ϑ j,p = θ j,p -θ j,p θ j,p = 1 √ T θ j,p ξ j,p + 1 T ξ j,p + 1 T ς j,T + σ Q -σ T T , with ς j,p = E Q ξ 2 j,p -σ Q and ξ j,p = ξ 2 j,p -E Q ξ 2 j,p . Setting M(λ) = 1 √ T p j=1 λ(j)θ j ξ j,p and L(λ) = p j=1 λ(j) (66)
and using the functions (61) through the penalty term (37), we rewrite (65) as

Err(λ) = J T (λ) + 2 σ Q -σ T T L(λ) + 2M (λ) + 2 T B 1,Q (λ) + 2 P T (λ) B 2,Q (ν(λ)) σ Q T + S 2 -δ P T (λ), (67)
where ν(λ) = λ/|λ|. Let λ 0 = (λ 0 (j)) 1≤j≤ p be a fixed sequence in Λ and λ be defined as in (39). Substituting λ 0 and λ in (67), we obtain

Err( λ) -Err(λ 0 ) = J T ( λ) -J T (λ 0 ) + 2 σ Q -σ T T L( ) + 2 T B 1,Q ( ) + 2M ( ) + 2 P T ( λ) B 2,Q ( ν) σ Q T -2 P T (λ 0 ) B 2,Q (ν 0 ) σ Q T -δ P T ( λ) + δ P T (λ 0 ), where = λ -λ 0 , ν = ν( λ) and ν 0 = ν(λ 0 ). Now, in view of the inequality 2|ab| ≤ δa 2 + δ -1 b 2 we get that 2 P T (λ) |B 2,Q (ν(λ))| σ Q T ≤ δP T (λ) + B 2 2,Q (ν(λ)) δσ Q T .
Then, taking into account that |L( )| ≤ L( λ) + L(λ) ≤ 2Λ * and using the definition (62) we get

Err( λ) ≤ Err(λ 0 ) + 2M( ) + 2L 1,Q T + 2B * 2,Q δσ Q T + 4Λ * | σ T -σ Q | T + 2δ P T (λ 0 ) , where B * 2,Q = sup λ∈Λ B 2 2,Q ((ν(λ)).
To estimate the second term in the right side of this inequality we set S x = p j=1 x(j)θ j,p ψ j,p for x = (x(j)) 1≤j≤p ∈ R p . Thanks to (23) we estimate the term M(x) for any x ∈ R p as

E Q M 2 (x) ≤ κ Q 1 T p j=1 x 2 (j)θ 2 j,p = κ Q S x 2 T . (68)
Setting here

Z * = sup x∈Λ 1 T M 2 (x)/ S x 2 and Λ 1 = Λ -λ 0 , we get 2|M(x)| ≤ δ S x 2 + Z * T δ . ( 69
)
The last term here can be estimated from above as

E Q Z * ≤ x∈Λ 1 T E Q M 2 (x) S x 2 ≤ x∈Λ 1 κ Q = κ Q m * ,
where m * = card(Λ). Moreover, note that, for any

x ∈ Λ 1 , S x 2 -S x 2 = p j=1 x 2 (j)(θ 2 j,p -θ 2 j,p ) ≤ -2M 1 (x), (70)
where M 1 (x) = T -1/2 p j=1 x 2 (j)θ 2 j,p ξ j,T . Taking into account now that, for any x ∈ Λ 1 , the components |x(j)| ≤ 1, we can estimate this term as in (68), i.e. E Q M 2 1 (x) ≤ κ Q S x 2 /T . Similarly to the previous reasoning setting

Z * 1 = sup x∈Λ 1 T M 2 1 (x)/ S x 2 , we get E Q Z * 1 ≤ κ Q m * .
Using the same type of arguments as in (69), we can derive

2|M 1 (x)| ≤ δ S x 2 + Z * 1 T δ . (71)
From here and (70), we get

S x 2 ≤ S x 2 1 -δ + Z * 1 T δ(1 -δ) (72)
for any 0 < δ < 1. Using this bound in (69) yields

2M (x) ≤ δ S x 2 1 -δ + Z * + Z * 1 T δ(1 -δ) . Taking into account that | S | 2 ≤ 2 (Err( λ) + Err(λ 0 )), we obtain 2M ( ) ≤ 2δ(Err( λ) + Err(λ 0 )) 1 -δ + Z * + Z * 1 T δ(1 -δ)
and, therefore,

Err( λ) ≤ 1 + δ 1 -3δ Err(λ 0 ) + Z * + Z * 1 T δ(1 -3δ) + 2L 1,Q T (1 -3δ) + 2B * 2,Q δ(1 -3δ)σ Q T + (4Λ * + 2) T (1 -3δ) | σ T -σ Q | + 2δ (1 -3δ) P T (λ 0 ).
Moreover, for 0 < δ < 1/6 we can rewrite this inequality as

Err( λ) ≤ 1 + δ 1 -3δ Err(λ 0 ) + 2(Z * + Z * 1 ) T δ + 4L 1,Q T + 4B * 2,Q δσ Q T + (8Λ * + 2)| σ T -σ Q | T + 2δ (1 -3δ) P T (λ 0 ).
Now, using the upper bound (63), we get

E Q B * 2,Q ≤ λ∈Λ E Q B 2 2,Q (ν(λ)) ≤ m * L 2,Q . Now, taking into account that Λ * ≥ 1, we get R Q ( S * , S) ≤ 1 + δ 1 -3δ R Q ( S λ 0 , S) + 4κ Q m * T δ + 4L 1,Q T + 4m * L 2,Q δσ Q T + 10Λ * E Q | σ T -σ Q | T + 2δ (1 -3δ) P T (λ 0 ).
By using the upper bound for P T (λ 0 ) in Lemma 4, we obtain that

R Q ( S * , S) ≤ 1 + 3δ 1 -3δ R Q ( S λ 0 , S) + 4κ Q m * T δ + 4L 1,Q T + 4m * L 2,Q δσ Q T + 10Λ * T E Q | σ T -σ Q | + 2δL 1,Q (1 -3δ)T .
Taking into account here that 1 -3δ ≥ 1/2 for 0 < δ < 1/6 and, moreover, that κ Q ≤ (1 + τ |ρ| * )σ Q and using the bounds ( 62) and ( 63) we obtain the inequality (41). Hence we obtain the desired result.

where M is given in (66). Setting x = (x j ) 1≤j≤p and x j = p -1/2 1 {[ √ T ] j p} , one can write the last term on the right hand side of (73) as

1 p p j=[ √ T ] ξ 2 j,p = 1 √ p B 2,Q (x ) + 1 p B 1,Q (x ) + (p -[ √ T ] + 1)σ Q p ,
where the functions B 1,Q and B 2,Q are defined in (61). Using Proposition 4 and Lemma 6 , we come to the following upper bound

E Q | σ T -σ Q | ≤ 16| Ṡ| 2 T [ √ T ]p + 2T p E Q |M(x )| + L 1,Q p + L 2,Q √ p + σ Q ([ √ T ] -1) p .
In the same way as in (68) through Lemma 6, we obtain

E Q |M (x )| ≤   κ Q T p j=[ √ T ] θ 2 j,p   1/2 ≤ 4(κ Q | Ṡ| 2 ) 1/2 [ √ T ] .
Taking into account that κ Q ≤ (1 + τ |ρ| * )σ Q and using the bounds ( 62) and ( 63) we obtain the inequality (42). Hence Proposition 1 holds true.

Proof of Theorem 2

This proof directly follows from Theorem 1 and Proposition 1.

Proof of Theorem 4

First, we denote by Q 0 the distribution in D[0, n] of the noise (6) with the parameter 1 = ς * , 2 = 0 and 3 = 0, i.e., the distribution for the "signal + white noise" model. So, we can estimate from below the robust risk R * T ( S T , S) ≥ R Q 0 ( S T , S). Now, Theorem 6.1 from Konev Pergamenshchikov (2009b) yields the bound (50). Hence we obtain the desired result.

Proof of Proposition 2

First, we note that in view of (31) one can represent the quadratic risk for the empiric norm • p defined in ( 25) as

E Q S λ 0 -S 2 p = 1 T p j=1 λ 2 0 (j) E Q ξ 2 j,p + Θ p ,
where Θ p = p j=1 θ j,p -λ 0 (j) θ j,p 2 . First, note that Now, from (29) we obtain that for any 0 < ε < 1

Θ p ≤ (1 + ε) Θ p + (1 + ε -1 ) p j=1 h 2 j,p , (75)
where Θ p = p j=1 (1 -λ 0 (j)) 2 θ 2 j,p . Moreover, in view of the definition (51)

Θ p = [ω 0 ]
j=ι 0

(1 -λ 0 (j)) 2 θ 2 j,p + (1 -λ 0 (j)) 2 θ 2 j + 4π 2 r(1 + ε -1 ) ω 3 0 p -2 .

Through Lemma 7 we have Θ 2,p ≤ (1 + ε) j≥[ω 0 ]+1 θ 2 j + (1 + ε -1 ) r p -2 . Hence, Θ p ≤ (1 + ε) Θ * + (1 + ε -1 ) 4π 2 rω 3 0 + r p -2 , where the first term Θ * = j≥ln T (1 -λ 0 (j)) 2 θ 2 j . Moreover, note that sup (1 -λ 0 (j)) 2 /a j ,

where the sequence (a j ) j≥1 is defined in (48). This leads to the inequality sup

Using now Lemma 5 and the condition H 5 ), we get the upper bound (52).

Hence we obtain the desired result.

Keywords

  Regression model • Generalized semi-Markov processes • fractional Poisson processes • Non-asymptotic estimation • Robust estimation • Model selection • Sharp oracle inequalities • Asymptotic efficiency • Adaptive estimation

H 6 )

 6 The parameters m * and Λ * defined in (32) can be functions of T , i.e. m * = m * (T ) and Λ * = Λ * (T ), such that for any b > 0 lim T →∞ T -b m * (T ) = 0 and lim T →∞ T -1/3-b Λ * (T ) = 0. Now, Theorem 2 implies the following oracle inequality.

7. 2

 2 Proof of Proposition 1 Let x = (x j ) 1≤j≤p with x j = 1

  ) + L 1,Q . where L * 1,T = sup Q∈Q T L 1,Q .Taking into account that υ T = T /σ * , we get sup

  := Θ 1,p + Θ 2,p , where ω 0 = ω α 0 = (τ k r 0 υ T ) 1/(2k+1) and r 0 = [r/ε] ε. Lemma 8 yields Θ 1,p ≤ (1 + ε) [ω 0 ] j=[ln T ]

  ≤ Ṡ 2 p -2 ≤ r p -2 .Moreover, W r,k ⊆ W r,2 for any k ≥ 2. From here and Lemma 9 ≤ r p -1 1 {k=1} + 3p -2 1 {k≥2} and, therefore, in view of the condition H 5 )

E

  2 j ≤ U T r . Using lim T →∞ r 0 = r, we get lim sup T →∞ U T ≤ π -2k (τ k r) -2k/(2k+1) , where the coefficient τ k is given in (74). This implies immediately that lim supT →∞ υ 2k/(2k+1) T sup S∈W r,k Θ p ≤ r 1/(2k+1) π 2k (τ k ) 2k/(2k+1) . Q S λ 0 -S 2 p ≤ l * .

Note that there is a very important particular case of the spherically symmetric vectors represented by Gaussian mixture distributions. The vector ζ = (ζ 1 , . . . , ζ d ) is called Gaussian mixture in R d if it has the spherically symmetric distribution with

  ). A random vector ζ = (ζ 1 , . . . , ζ d ) is called spherically symmetric if its density in R d has the form g(| • | 2 ) for some nonnegative function g. Here the prime denotes the transposition.

  (ζ j ) j≥ 1 are i.i.d. random variables satisfying condition C 1 ); for some d > 1 the random vector (ζ 1 , . . . , ζ d ) that has a spherically symmetric distribution in R d , with E ζ 2 1 = 1, E ζ 4 1 < ∞ and the random variables (ζ j ) j> d are independent and satisfying condition C 1 ); for any d ≥ 1 a random vector (ζ 1 , . . . , ζ d ) is a Gaussian mixture with mixture variable s for which E s 2 = 1 and E s 4 < ∞.

This research was supported by RSF, project no 20-61-47043 (National Research Tomsk State University, Russia)

Appendix

A.1 Property of the penalty term Lemma 4 For any n ≥ 1 and λ ∈ Λ,

where the coefficient P T (λ) is defined in (66) and L 1,Q is defined in (62).

Proof. From ( 30) and ( 33) we obtain

Hence we obtain the result.

A.2 Properties of the Fourier coefficients

Lemma 5 Let f be an absolutely continuous function,

, where c j are some constants. Then, for any ε > 0, the function ∆ = f -g satisfies the following inequalities

Lemma 6 Let the function S(t) in (3) be absolutly continuous and have an absolutely integrable derivative. Then the coefficients (θ j,p ) 1 j p defined in (29) satisfy the inequalities |θ 1,p | S 1 and max 2 j p j|θ j,p | 2 √ 2| Ṡ| 1 .

Lemma 7 For any p ≥ 2, 1 ≤ N ≤ p and r > 0, the coefficients (θ j,p ) 1≤j≤p of functions S from the class W r,1 satisfy, for any ε > 0, the following inequality

Lemma 8 For any p ≥ 2 and r > 0, the coefficients (θ j,p ) 1≤j≤p of functions S satisfy the inequality max 1≤j≤p sup S∈W r,1 |θ j,p -θ j | -2π √ r j p -1 ≤ 0.

Lemma 9 For any p ≥ 2 and r > 0 the correction coefficients from (29) satisfy the inequality sup S∈W r,2 p j=1 h 2 j,p ≤ 3r p -2 .

Lemmas 5 -9 are proven in [START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF].


Ken Ono
email: ken.ono691@virginia.edu

Robert Schneider
email: robert.schneider@uga.edu

Ian Wagner
email: ian.c.wagner@vanderbilt.edu

Partition-theoretic formulas for arithmetic densities, II

Keywords: integer partitions, arithmetic density, q-series, q-binomial theorem 2010 Mathematics Subject Classification. 05A17, 11P82

published or not. The documents may come

Introduction and statement of results

It is well-known that the Möbius function satisfies

-lim N →∞ N n=2 µ(n) n = 1. (1.1)
This fact can be used to obtain formulas for arithmetic densities for suitable subsets of the positive integers. A celebrated example of this idea was obtained by Alladi [START_REF] Alladi | Duality between prime factors and an application to the prime number theorem for arithmetic progressions[END_REF], who proved for r, t, ∈ N that if gcd(r, t) = 1, then

- n≥2 p min (n)≡r (mod t) µ(n) n = 1 ϕ(t)
.

(1.2)

Here ϕ(t) is Euler's phi function, and p min (n) is the smallest prime factor of n.

Over the last few years, (1.2) has been generalized in various ways in works by Dawsey [START_REF] Dawsey | A new formula for Chebotarev densities[END_REF], Sweeting and Woo [START_REF] Sweeting | Formulas for Chebotarev densities of Galois extensions of number fields[END_REF], Kural, McDonald and Sah [KMS], and Wang [START_REF] Wang | An analogue of a formula for Chebotarev densities[END_REF][START_REF] Wang | Ramanujan sum and Chebotarev densities[END_REF]. In a recent preprint [START_REF] Wang | Analogues of Alladi's formula[END_REF], Wang obtained a beautiful generalization of this phenomenon which makes use of Dirichlet convolutions. If a : N → C is an arithmetic function satisfying a(1) = 1 and In earlier work the authors obtained a partition-theoretic analogue of formulas such as (1.2) using natural analogies between the multiplicative structure of the integers and the additive structure of partitions. The purpose of this note is to show that an analogue also exists for (1.3), as a consequence of a general theorem offering infinitely many formulas (see Theorem 1.3).

We first recall fundamental features of these analogies. A partition is a finite non-increasing sequence of positive integers, say λ = (λ 1 , λ 2 , ..., λ (λ) ), where (λ) denotes the length (number of parts) of λ. The size of λ is |λ| := λ 1 + λ 2 + • • • + λ (λ) (sum of parts) and the norm of the partition is N (λ) = λ 1 • λ 2 • • • λ (λ) (product of parts). 1 Furthermore, we let sm(λ) := λ (λ) denote the smallest part of λ (resp. lg(λ) := λ 1 the largest part of λ).

We will use the partition Möbius function µ P (λ) := 0 if λ has repeated parts, (-1) (λ) otherwise.

(1.4)

Notice that µ P (λ) = 0 if λ has any repeated parts, which is analogous to the vanishing of µ(n) for integers n which are not square-free. In particular, the different parts in a partition λ play the role of prime factors of n in this analogy. We define µ * P (λ) := -µ P (λ) as in Dawsey's theorem, to eliminate minus signs in our formulas. Finally, we recall the usual q-Pochhammer symbol (a; q) n := n-1 k=0 (1 -aq k ) and (a; q) ∞ := lim n→∞ (a; q) n for a, q ∈ C, |q| < 1.

To complete the analogy, also recall the partition phi function defined in [START_REF] Schneider | Arithmetic of partitions and the q-bracket operator[END_REF], an analogue of Euler's totient function,

ϕ P (λ) := N (λ) • k∈λ without repetition 1 -k -1 , where "k ∈ λ" means k ∈ N is a part of λ ∈ P.
Table 1 offers some of the identifications making up this multiplicative-additive analogy.

Natural number n Partition λ Prime factors of n

Parts of λ Square-free integers Partitions into distinct parts

µ(n) µ P (λ) ϕ(n) ϕ P (λ) p min (n) sm(λ) p max (n) lg(λ) n -s q |λ| ζ(s) -1 (q; q) ∞ s → 1 q → 1
Table 1: Analogies between arithmetic and partition theory Remark 1.1. We note that other, different multiplicative-additive analogies exist, such as those drawn in [START_REF] Schneider | Analysis and combinatorics of partition zeta functions[END_REF] between partitions of length m and those of size m, in which the duality instead correlates ζ(s) to the geometric series q(1 -q) -1 , with m -s in bijection with q m . To describe our results, let S be a subset of the positive integers with arithmetic density

d S := lim N →∞ #{n ∈ S : n ≤ N } N .
1 The norm is referred to as the "integer" of λ and written n λ in some earlier works [START_REF] Ono | Explorations in the theory of partition zeta functions, Exploring the Riemann Zeta Function, 190 years from Riemann's Birth[END_REF][START_REF] Schneider | Partition zeta functions[END_REF].

As noted above, we let µ * P (λ) := -µ P (λ). In [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF] the function

F S (q) := λ∈P sm(λ)∈S µ * P (λ)q |λ| (1.5)
was defined, and it was shown, for suitable S and associated roots of unity ζ, that lim q→ζ F S (q) = d S .

(1.6)

This theorem is a partition-theoretic analogue of (1.2).

Remark 1.2. We note that throughout this paper, "q → ζ" with ζ a root of unity (including the case ζ = 1) denotes that q approaches the point on the unit circle from within the unit disk.

To state the partition analogue of Wang's theorem requires some further notation for partition multiplication, division, and subpartitions that are analogous to partition divisors. For two partitions λ, δ ∈ P, we define δλ to be the partition obtained by concatenating the parts of λ to δ (and then reordering by size by notational convention). We write δ|λ if all of the parts of δ appear in the partition λ with greater than or equal multiplicities and define the subpartition λ/δ as the partition obtained by removing those parts, counting multiplicities, from λ.

For two functions f and g on partitions, we define their partition Dirichlet convolution by

(f * g)(λ) := δ|λ f (δ)g(λ/δ), (1.7)
where the sum is taken over the subpartitions δ of λ (including the empty partition δ = ∅).

To state our results succinctly, we define an auxiliary series that effectively serves to identify different classes of density results. For |q| < 1, a(λ) a function on P, and n ≥ 1, let

A n (q) := a ((n)) + sm(γ)≥ * n [a(γ • (n)) -a(γ)] q |γ| ,
(1.8) with (n) ∈ P the partition of n with one part, γ • (n) ∈ P the partition formed by adjoining part n to partition γ, and with ≥ * denoting > if a(λ) = 0 at partitions with any part repeated, and denoting ≥ otherwise. We introduce one final piece of terminology: throughout this paper, the formulas we prove hold true for "nice" subsets S ⊆ N amenable to q-series density calculations, which we call q-commensurate subsets (see Section 3 below for the precise definition).

Theorem 1.3. Let S ⊆ N be a q-commensurate subset, and a(λ) a function on P such that a(∅) = 1.

If lim q→1 A n (q) = 1 when n ∈ S, then lim q→1 sm(λ)∈S (µ P * a)(λ)q |λ| = 0. If lim q→1 A n (q) = 1 + f (n) for f (n) an arithmetic function nonzero when n ∈ S, then lim q→1 sm(λ)∈S (µ P * a)(λ) f (sm(λ)) q |λ| = d S .
If we set f (n) = n, Theorem 1.3 gives the partition analogue of (1.3). Immediately we have a nice generalization of formula (1.6) with ζ = 1.

Corollary 1.4. For S ⊆ N a q-commensurate subset, if lim q→1 A n (q) = 0 when n ∈ S, then

-lim q→1 sm(λ)∈S (µ P * a)(λ)q |λ| = d S .
Remark 1.5. Set a(λ) = δ|λ µ P (δ) = 1 if λ = ∅ and = 0 otherwise; one computes A n (q) = 0 for all n ≥ 1. Then the theorem reduces to the main result of [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF], viz. lim q→1 F S (q) = d S .

Furthermore, for an arithmetic function a(n), Wang defines b(n) := d|n (µ * a)(d) d ϕ(d) with ϕ(k) the classical Euler phi function, and in equation (36) of [START_REF] Wang | Analogues of Alladi's formula[END_REF] uses Möbius inversion to show

(µ * a)(n) ϕ(n) = (µ * b)(n) n .
(1.9)

This along with the main theorem (1.3) allows Wang to prove beautiful formulas like

-lim N →∞ 2≤n≤N p min (n)∈S (µ * a)(n) ϕ(n) = d S .
(1.10)

Replacing a, b with functions on partitions, Theorem 1.3 yields an analogue of these equations.

Corollary 1.6. Let S ⊆ N be a q-commensurate subset, a(λ) a function on P such that a(∅) = 1, f and g functions on Z ≥0 such that f (n) = 0, g(n) nonzero when n ∈ S, and

b(λ) = b a,f,g (λ) := δ|λ (µ P * a)(δ) g (sm(δ)) f (sm(δ)) . If lim q→1 A n (q) = 1 + f (n) for n ∈ S, then lim q→1 sm(λ)∈S (µ P * b)(λ) g (sm(λ)) q |λ| = d S .
A version of Corollary 1.6 gives an analogue to Wang's identity (1.10).

Corollary 1.7. Let S ⊆ N be a q-commensurate subset, and take a(λ), b(λ) = b a,f,g (λ) as in Corollary 1.6. In analogy with (1.8), define

B n (q) := b ((n)) + sm(γ)≥ * n [b(γ • (n)) -b(γ)] q |γ| . Let f (n) = -ϕ(n), g(n) = -1, so that b = b a,-ϕ,-1 . If lim q→1 B n (q) = 0 for n ∈ S, then -lim q→1 sm(λ)∈S (µ P * a)(λ) ϕ (sm(λ)) q |λ| = d S ,
where ϕ(n) is the classical phi function.

We prove these limiting formulas in Section 4.

Remark 1.8. As noted in Section 3 below, by a Tauberian theorem of Frobenius [Fr1880], the formulas in this paper hold for every subset S ⊆ N if one lets q → 1 -strictly radially.

Conceptual background 2.A. Partition analogue of multiplicative number theory

Before proceeding to prove the propositions in Section 1 let us discuss the multiplicative-additive confluence that provides the context in which results like these -displaying a fusion of partitiontheoretic objects with familiar-looking forms from classical multiplicative number theory -arise naturally.

In the preceding section we introduced the multiplicative partition norm N (λ), the partition product (concatenation), partition division (deleting parts) and subpartition sums that act like sums over divisors, as well as partition analogues of classical arithmetic functions and Dirichlet convolution; this multiplicative theory of additive partitions was introduced in [Sc16, Sc17, Sc18] and subsequent work [START_REF] Ono | Explorations in the theory of partition zeta functions, Exploring the Riemann Zeta Function, 190 years from Riemann's Birth[END_REF][START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF][START_REF] Schneider | Analysis and combinatorics of partition zeta functions[END_REF]. Some of these analogies, pertinent to this paper, are captured in Table 1.

These partition-theoretic structures behave almost identically to their classical counterparts. For instance, one can prove familiar-looking "divisor sum" identities like

δ|λ µ P (δ) = 1 if λ = ∅, 0 otherwise, δ|λ ϕ P (δ) = N (λ);
the partition Möbius inversion formula which is obviously related to partition convolution defined in (1.7).2 We note two useful consequences of (2.12) from [START_REF] Schneider | Arithmetic of partitions and the q-bracket operator[END_REF], setting b = µ P :

A(λ) = δ|λ a(δ) ⇐⇒ a(λ) = δ|λ A(δ)µ P (λ/δ), (2.11
(q; q) -1 ∞ λ∈P a(λ)q |λ| = λ∈P q |λ| δ|λ a(δ), (q; q) ∞ λ∈P a(λ)q |λ| = λ∈P q |λ| δ|λ a(δ)µ P (λ/δ).
These are q-series analogues of the classical Dirichlet series identities (see Table 1)

ζ(s) -1 ∞ n=1 f (n)n -s = ∞ n=1 n -s d|n f (d), ζ(s) ∞ n=1 f (n)n -s = ∞ n=1 n -s d|n f (d)µ(n/d).
This multiplicative-additive theory also contains broad classes of partition zeta functions

ζ P (s) := λ∈P N (λ) -s (Re(s) > 1)
where P P, that yield beautiful evaluations [START_REF] Schneider | Partition zeta functions[END_REF] and interesting analytic information [START_REF] Ono | Explorations in the theory of partition zeta functions, Exploring the Riemann Zeta Function, 190 years from Riemann's Birth[END_REF][START_REF] Schneider | Analysis and combinatorics of partition zeta functions[END_REF]. There also exist partition Dirichlet series [ORS17, Sc18] that enjoy identities generalizing classical Dirichlet series relations, e.g. if P S denotes partitions with all parts in any S ⊂ N,

λ∈P S µ P (λ)N (λ) -s = 1 ζ P S (s) (Re(s) > 1), λ∈P S ϕ P (λ)N (λ) -s = ζ P S (s -1) ζ P S (s) (Re(s) > 2).
Taking S = P regains the classical cases. While this is indeed looking a lot like multiplicative number theory, the same superstructure encompasses q-hypergeometric series, partition bijections and other aspects of additive number theory, and recovers classical theorems in those areas as well (see [START_REF] Schneider | Eulerian Series, Zeta Functions and the Arithmetic of Partitions[END_REF] for further reading).

3. Computing arithmetic densities with q-series 3.A. The q-density statistic

For |q| < 1, we recall that the function F S (q) from (1.5) has two natural representations, one involving sm(λ) and one involving lg(λ), the smallest and largest parts of partition λ, respectively. Namely, we have

F S (q) := sm(λ)∈S
µ * P (λ)q |λ| = lg(λ)∈S q |λ| λ∈P q |λ| = (q; q) ∞ n∈S q n (q; q) n , (3.13) noting sm(∅) = lg(∅) := 0. We proved in [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF] that lim q→1 F S (q) = d S for special subsets S ⊆ N.

The story arc represented in the works of Alladi [START_REF] Alladi | Duality between prime factors and an application to the prime number theorem for arithmetic progressions[END_REF], Dawsey [START_REF] Dawsey | A new formula for Chebotarev densities[END_REF], Wang [Wa1, Wa2, Wa3], et al., together with our own work in [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF], stems from the observation that (1.1) can be interpreted as a statement about arithmetic density. Now, this statement itself can be interpreted via L'Hospital's rule and Lambert series as a statement about q-series, since

n≥2 µ(n) n = lim q→1 (1 -q)   n≥1 µ(n)q n 1 -q n - µ(1)q 1 -q   = lim q→1 (1 -q) q - q 1 -q = -1,
assuming the implicit order-of-limits swap with "lim q→1 " and the infinite series is valid. The classical q-binomial theorem was at the heart of our results in [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF]. In order to prove the main results of this paper involving less familiar-looking forms, we sketch a theory of q-series computations of arithmetic density from first principles, based on a simple statistic. Definition 3.1. For |q| < 1 we define the q-density of S ⊆ N by the quotient

d S (q) := n∈S q n n≥0 q n = (1 -q) n∈S q n . (3.14)
The q-density is a geometric series analogue of (3.13).3 Of course, from both the shape of d S (q) and known instances of F S (q), one wants to see the limiting behavior lim q→1 n∈S q n n≥0 q n = lim

N →∞ n∈S(N ) 1 0≤n<N 1 =: d S , (3.15)
where S(N ) = {n ∈ S : n ≤ N }. It is a consequence of a Tauberian theorem due to Frobenius with a converse by Hardy-Littlewood (see [Fr1880, Ko]), that in fact (3.15) holds for every subset S ⊆ N if q → 1 radially. 4 In [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF], we prove arithmetic density formulas using other paths, and approaching other roots of unity; we do not rigorously prove general limiting properties here (certainly such a study would be useful), but side-step the issue somewhat circularly by defining it to be a property of a subset of N that the limit computations we undertake are valid.

Definition 3.2. We define a subset S ⊆ N to be q-commensurate if lim q→1 d S (q) = d S is approached sufficiently rapidly for the asymptotic proofs we employ here to hold.

Remark 3.3. In full generality, a subset S being q-commensurate is equivalent to one's being able, for that subset, to interchange order of limits in the calculation d S := lim N →∞ lim q→1 n∈S(N ) q n 0≤n<N q n , and to having all implicit error terms in the following asymptotic equalities of lower order than the main terms, so the errors vanish when multiplied by (q; q) ∞ as q → 1.

Immediately we see that S = N is q-commensurate since d N (q) = q → 1 = d N . The analogy between d S (q) and F S (q) holds quite closely with respect to density.

Theorem 3.4. Let r ∈ Z, t ∈ N. For subsets S r,t := {n ≥ 0 : n ≡ r (mod t)} we have that lim q→1 d Sr,t (q) = d Sr,t = 1 t .

Proof. The claimed limit follows from geometric series and L'Hospital's rule:

lim q→1 d Sr,t (q) = lim q→1 (1 -q) k≥0 q kt+r = lim q→1 q r (1 -q) 1 -q t = 1 t .
Theorem 3.4 is a geometric series version of Theorem 1.3 in [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF]. There is also a q-density analogue of Corollary 1.5 in [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF].

Corollary 3.5. For subsets S (k) fr ⊂ N of kth power-free integers, k ≥ 2, we have that

lim q→1 d S (k) fr (q) = d S (k) fr = 1 ζ(k) .
Proof. The steps of the proof of Cor. 1.5 in [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF] hold identically here, only one replaces F S r,M (q) with d S r,M (q), then uses Theorem 3.4 in place of Th. 1.3 of [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF].

Thus the subsets S r,t and S (k) fr are q-commensurate. Table 2 illustrates the k = 2 (square-free) case of Corollary 3.5 as q → 1 radially.

3.B. Further q-series density formulas

The q-density statistic can be used as a building block for further arithmetic density relations. 6 For S ⊆ N a q-commensurate subset7 , as q → 1 it follows from (3.15) that n∈S

q n ∼ d S • q 1 -q . (3.16)
Because geometric series are central components of partition generating functions and q-series, the q-density may be embedded in diverse partition relations. For example, one can prove the central claims of [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF] (with a sufficient condition) by applying generating function methods to (3.16).

Theorem 3.6. For S ⊆ N a q-commensurate subset, we have

lim q→1 F S (q) = -lim q→1 n∈S k≥1 (-1) k q nk+ k(k-1) 2 (q; q) k-1 = d S .
Proof. Take q → q k in (3.16). Multiply through by (-1) k q k(k-1) 2

(q; q) -1 k-1 and sum both sides over k ≥ 1, swapping order of summation on the left to give n∈S k≥1 (-1) k q nk+ k(k-1) 2 (q; q) k-1 .

(3.17)

For each k ≥ 1, the factor (q; q) -1 k-1 generates partitions with largest part strictly < k. The factor q nk adjoins a largest part k with multiplicity n ∈ S to each partition. The q k(k-1)/2 = q 1+2+3+...+(k-1) factor guarantees at least one part of each size < k. Thus (3.17) is the generating function for partitions γ with largest part having multiplicity in S, and with every natural number < lg(γ) appearing as a part, weighted by (-1) lg(γ) = (-1) k . Under conjugation, this set of partitions γ maps to partitions λ into distinct parts with smallest part sm(λ) ∈ S, weighted by µ P (λ) = (-1) (λ) = (-1) k which is nonzero since λ has no repeated part. Multiplying by -1 gives µ * P (λ) := -µ P (λ); thus

- n∈S k≥1 (-1) k q nk+ k(k-1) 2 (q; q) k-1 = sm(λ)∈S µ * P (λ)q |λ| = F S (q).
Manipulating the right-hand side of (3.16) accordingly, we then have

- n∈S k≥1 (-1) k q nk+ k(k-1) 2 (q; q) k-1 ∼ -d S • k≥1 (-1) k q k(k+1) 2 (q; q) k = -d S • ((q; q) ∞ -1) ,
using an identity of Euler in the last step, which clearly approaches d S as q → 1.

We don't necessarily need to restrict smallest parts to S to give formulas for d S . For instance, by similar methods, an analogous formula extends to the "largest parts" case.

Theorem 3.7. For S ⊆ N a q-commensurate subset, we have

lim q→1 lg(λ)∈S µ * P (λ)q |λ| = d S .
Proof. Take q → q k in (3.16). Multiply both sides by (q; q) -1 k-1 , sum over k ≥ 1, then swap order of summation and make the change of indices k → k + 1 on the left side, to give

n∈S q n k≥0 q nk (q; q) k ∼ d S • k≥1 q k (q; q) k . (3.18)
By the q-binomial theorem, the inner sum over k ≥ 0 is equal to (q n ; q) -1 ∞ , and the summation on the right equals (q; q) -1 ∞ -1. Then multiplying both sides of (3.18) by (q; q) ∞ implies by well-known generating function arguments that, as q → 1, we have

n∈S q n (q; q) n-1 = lg(λ)∈S µ * P (λ)q |λ| ∼ d S . (3.19)
Remark 3.8. Taken together, Theorems 3.6 and 3.7 present an "Alladi duality" connecting relations over minimal and maximal elements of multisets, as exemplified in [START_REF] Alladi | Duality between prime factors and an application to the prime number theorem for arithmetic progressions[END_REF][START_REF] Alladi | A duality between the largest and smallest prime factors via the Möbius function and arithmetical consequences[END_REF][START_REF] Wang | Analogues of Alladi's formula[END_REF]. Whereas in [START_REF] Alladi | Duality between prime factors and an application to the prime number theorem for arithmetic progressions[END_REF] the smallest and largest prime factors of n induce a duality, here the duality is between the smallest and largest parts of λ.

Proof of Theorem 1.3 and its corollaries 4.A. Partition convolution identities

Now we turn our attention toward proving the partition convolution identities analogous to formulas of Wang [START_REF] Wang | Analogues of Alladi's formula[END_REF]. It is an instance of (2.12) for arbitrary partition-theoretic functions a(λ), b(λ)

that λ∈P a(λ)q |λ| λ∈P b(λ)q |λ| = λ∈P (a * b)(λ)q |λ| = n≥0 sm(λ)=n (a * b)(λ)q |λ| , (4.20)
where we now take * to be partition-theoretic convolution (which is apparent from the context). We record a useful formula for the inner sum sm(λ)=n of the right-hand double series. This will serve as another building block for further arithmetic density formulas.

Lemma 4.1. For a(λ), b(λ) functions on P, we have where ≥ * denotes > if the associated summands vanish at partitions with any part repeated, and denotes ≥ otherwise.

sm(λ)=n (a * b)(λ)q |λ| =   sm(γ)=n a(γ)q |γ|     b(∅) + sm(γ)≥ * n b(γ)q |γ|   +   a(∅) + sm(γ)≥ * n a(γ)q |γ|     sm(γ)=n b(γ)q |γ|   ,
Proof. The right-hand side of Lemma 4.1 generates the partitions λ with smallest part n on the left. That the summands on the left should have the form q |λ| δ|λ a(δ)b(λ/δ) = δδ =λ a(δ)q |δ| • b(δ )q |δ | is obvious from standard partition generating function ideas, e.g. (4.20). Now consider λ = δδ on the left, choosing δ from a sum with coefficients a(γ) on the right-hand side, and δ from a sum with coefficients b(γ). If subpartition δ contains the copy of part n satisfying the "sm(λ) = n" condition, then λ is generated by the first term "( sm(γ)=n )(b + sm(γ)≥ * n )" on the right: δ comes from the sm(γ)=n factor, and δ comes from the b + sm(γ)≥ * n factor. If, on the other hand, δ contains the designated smallest part n, then λ is generated by the second term "(a + sm(γ)≥ * n )( sm(γ)=n )" on the right: δ comes from a + sm(γ)≥ * n and δ comes from the sm(γ)=n factor.

We wish to find formulas with coefficients of the form µ P * a, to yield Theorem 1.3. Now, for |q| < 1, recall the q-bracket a q := λ∈P a(λ)q |λ| λ∈P q |λ| = (q; q) ∞ λ∈P a(λ)q |λ| of Bloch and Okounkov.

The q-bracket can be interpreted as the expected value of a(λ) over all partitions, given certain hypotheses from statistical physics; it is connected to modular, quasi-modular and p-adic modular forms (see [BlOk00, BOW, GJT16, Za15]). Then (4.20) gives λ∈P

(µ P * a)(λ)q |λ| = a q , (4.21)
which of course leads to further partition convolution relations from q-bracket examples in the literature. Taken over all partitions λ, we see µ P * a induces a type of averaging phenomenon, as well as a connection to the theory of modular forms.

To restrict smallest parts to the subset S, we require a specialization of Lemma 4.1.

Lemma 4.2. For a(λ) a function on P, we have

sm(λ)=n (µ P * a)(λ)q |λ| = A n (q) • q n (q n+1 ; q) ∞ ,
where A n (q) := A n (q) -a(∅) with A n (q) as in (1.8).

Proof

(µ P * a)(λ)q |λ| = -q n (q n+1 ; q) ∞   a(∅) + sm(γ)≥ * n a(γ)q |γ|   + (q n+1 ; q) ∞   sm(γ)=n a(γ)q |γ|   .
Factor q n out of the sm(γ)=n factor on the right; the summands undergo the transformation a γ)q |γ| → a(γ • (n) q |γ| . Then a little arithmetic gives sm(λ)=n

(µ P * a)(λ)q |λ| = q n (q n+1 ; q) ∞   a ((n)) -a(∅) + sm(γ)≥ * n [a(γ • (n)) -a(γ)] q |γ|   .
The parenthetical term on the right is exactly A n (q), completing the proof. This lemma works out nicely when the choice of a(λ) allows one to factor out aspects of A n (q), or simplify the sum otherwise. Now the following general lemma, wrapping together the preceding formulas, will lead to the proof of Theorem 1.3 as q → 1.

Lemma 4.3. For a(λ) a partition-theoretic function, f (n) an arithmetic function, we have

λ =∅ (µ P * a)(λ)f (sm(λ)) q |λ| = (q; q) ∞ n≥1 f (n) A n (q)q n (q; q) n ,
where A n (q) := A n (q) -a(∅) with A n (q) as in (1.8).

Proof. Multiply Lemma 4.2 through by f (n), and sum over n ≥ 1, noting for each index n the partitions generated on the left-hand side have smallest part sm(λ) = n. Swapping order of summation on the left, then, takes f (n) → f (sm(λ)). On the right-hand side, use the factorization (q n+1 ; q) ∞ = (q;q)∞ (q;q)n to give the Lemma 4.3.

4.B.

Proof of Theorem 1.3 and Corollaries 1.4 and 1.6

We now have all the ingredients to prove Theorem 1.3 and its corollaries.

Proof of Theorem 1.3. Consider the identity in Lemma 4.3 above. By hypothesis we assume a(∅) = 1.8 Letting χ S (n) denote the indicator function for S ⊆ N, make the substitution

f (n) → χ S (n) f (n)
(hence the condition that f not vanish on S in the denominator) to give sm(λ)∈S

(µ P * a)(λ) f (sm(λ)) q |λ| = (q; q) ∞ n∈S A n (q)q n f (n) • (q; q) n . (4.22)
As q → 1, if A n (q) → 1 for n ∈ S, then A n (q) = A n (q) -1 → 0 in the summands on the right. Clearly the right side vanishes as q → 1, which proves the first identity of the theorem. Similarly, if A n (q) → 1 + f (n) for n ∈ S, then A n (q) → f (n). In the limit, An(q) f (n) → 1 and the right-hand side of (4.22) asymptotically equals F S (q), which approaches d S by Theorem 3.6. This is the content of the second identity and its analytic condition.

Remark 4.4. The idea is to set f (n) = lim q→1 A n (q) for n ∈ S, so the right-hand side becomes F S (q) in the limit.

Proof of Corollary 1.4. We have by hypothesis A n (q) = A n (q) + 1 → 0, which is equivalent to having lim q→1 A n (q) = -1. Then setting f (k) = -1 in (4.22) gives the corollary as q → 1. Divide through by g (sm(λ)) (thus the condition that it also is non-vanishing), multiply both sides by q |λ| , sum over partitions λ with sm(λ) ∈ S, then compare the result with the A n (q) → 1 + f (n) case of Theorem 1.3, to see lim

q→1 sm(λ)∈S (µ P * b)(λ) g (sm(λ)) q |λ| = lim q→1 sm(λ)∈S (µ P * a)(λ) f (sm(λ)) q |λ| = d S .
(4.24)

Proof of Corollary 1.7. With b(λ) = b a,-ϕ,-1 (λ), B n (q) as defined in Corollary 1.7, if B n (q) → 0 on S, the left-hand side of the resulting equation from (4.24) is the instance of Corollary 1.4 where a is replaced by b = b a,-ϕ,-1 . The right-hand equality of (4.24) then yields Corollary 1.7.

Further examples

In this section we give further examples of applications of the formulas and methods developed above. We note in passing for any q-commensurate subset S, that L'Hospital's rule yields infinitely many formulas for lim q→1 d S (q) by taking derivatives of the numerator and denominator of (3.14). Taking antiderivatives instead, then applying L'Hospital's rule, yields another arithmetic density relation for q-commensurate subset S:

-

lim q→1 1 log(1 -q) n∈S q n n = d S , (5.25)
where log z denotes the principle branch of the complex logarithm function for z ∈ C. The asymptotic (3.16) is suggestive of further, less trivial formulas to compute d S .

Example 5.1. Let µ(k) denote the classical Möbius function, and let f (k) be an arithmetic function that is not identically zero. For S ⊆ N a q-commensurate subset, we have

lim q→1 n∈S k≥1 (µ * f )(k)q nk k≥1 f (k)q k = d S ,
where * denotes classical Dirichlet convolution.

Proof. Take q → q k in (3.16). Multiply both sides of the asymptotic equality by (µ * f )(k) = d|k µ(d)f (k/d), sum over k ≥ 1, and (noting the series are absolutely convergent) swap order of summation on the left. Then as q → 1, well-known relations involving classical Lambert series and Dirichlet convolution give n∈S k≥1

(µ * f )(k)q nk ∼ d S • k≥1 (µ * f )(k)q k 1 -q k = d S • k≥1 f (k)q k .
(5.26)

Dividing through by k≥1 f (k)q k completes the proof.

Setting f (k) := 1 for all k ≥ 1, then Example 5.1 reduces to (3.15). Setting f (k) := σ 0 (k), the classical divisor function such that (µ * σ 0 )(k) = 1, yields a Lambert series calculation.

Example 5.2. For S ⊆ N a q-commensurate subset, we have

lim q→1 n∈S q n 1-q n n≥1 q n 1-q n = d S .
Setting f (1) := 1, f (k) := 0 for k > 1, viz. the function f (k) := d|k µ(d), gives another density formula that resembles a power series analogue of the first sum in (3.13).

Example 5.3. For S ⊆ N a q-commensurate subset, we have

lim q→1 n∈S k≥1 µ(k)q nk = d S .
Setting f (n) equal to the partition function p(n) in Example 5.1, gives an example that looks more like the formulas in [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF] and the present work.

Example 5.4. Let p(n) denote the partition function. For S ⊆ N a q-commensurate subset, we have

lim q→1 (q; q) ∞ n∈S k≥1 (µ * p)(k)q nk = d S .
Beyond giving analogues of classical convolution theorems as in Section 2. above, and giving formulas for expected value and arithmetic density, partition-theoretic convolution leads to other interesting q-series relations. Here is a nice consequence of Lemma 4.1, making another connection between partition convolution and modular forms.

Example 5.5. For z ∈ C, z = 0, set z (λ) := z (λ) , z -(λ) := z -(λ) . Recall the rank generating function for strongly unimodal sequences, U (z, q) := n≥0 q n+1 (-zq; q) n (-z -1 q; q) n . Then λ∈P (z * z -)(λ)q |λ| = 1 + z + z -1 (zq; q) ∞ (z -1 q; q) ∞ U (-z, q).

Remark 5.6. We note in Example 5.5 a close relation to the crank generating function of Andrews-Garvan, C(z, q) = (q;q)∞ (zq;q)∞(z -1 q;q)∞ (see [START_REF] Andrews | Dyson's crank of a partition[END_REF]), viz.

λ =∅ (z * z -)(λ)q |λ| = (z + z -1 )C(z, q)U (-z, q) (q; q) ∞ .

(5.27)

Proof. Take a = z , b = z -in Lemma 4.1 to yield sm(λ)=n (z * z -)(λ)q |λ| = zq n (zq n ; q) ∞ • 1 (z -1 q n ; q) ∞ + 1 (zq n ; q) ∞ • z -1 q n (z -1 q n ; q) ∞ = z + z -1 (zq; q) ∞ (z -1 q; q) ∞ • q n (zq; q) n-1 (z -1 q; q) n-1 .

Summing over n ≥ 1, then adding 1 to both sides of the equation to adjoin the λ = ∅ term on the left, and comparing with the definition of U (z, q), completes the proof.

The rank generating function for strongly unimodal sequences is connected to mock modular and quantum modular forms (see e.g. [START_REF] Folsom | Mock theta functions and quantum modular forms[END_REF]). Example 5.5 itself, equal to (zq; q) -1 ∞ (z -1 q; q) -1 ∞ by (4.20), is modular for z a root of unity, up to multiplication by a rational power of q.

One may obtain different forms of partition convolution identities for d S from those in Theorem 1.3 as consequences of Lemma 4.3, such as the following pair of identities.

Example 5.7. For S ⊆ N a q-commensurate subset, a(λ) = (λ) (partition length) and σ 0 (n) := d|n 1 as usual, we have lim q→1 sm(λ)∈S (µ P * )(λ)q |λ| n≥1 σ 0 (n)q n = d S .

For S ⊆ N a q-commensurate subset, a(λ) = sz(λ) := |λ| (partition size) and σ 1 (n) := d|n d, we have lim q→1 sm(λ)∈S (µ P * sz)(λ)q |λ| n≥1 σ 1 (n)q n = d S .

Remark 5.9. Note from (5.31) and from the fact q

(1-q) 2 = k≥1 ϕ(k)q k 1-q k = k≥1 kq k , that lim q→1 n∈S nq n n≥1 nq n = d S ,

(5.32) which can also be found by applying L'Hospital's rule to (3.14) as q → 1, as noted above.

We anticipate that formulas similar to those proved here will hold for sums over partitions with parts restricted in other ways to subsets S such as in Corollary 3.7, as well as for q → ζ a root of unity not equal to one, along the lines of those in [START_REF] Ono | Partition-theoretic formulas for arithmetic densities, Analytic Number Theory, modular forms and q-hypergeometric series[END_REF]; but pursuing this was outside of our present aims. Using limiting techniques such as the Hardy-Ramanujan circle method [HR1918], techniques of Watson [START_REF] Watson | The mock theta functions (2)[END_REF] and other methods from the universe of q-series and modular forms -as well as limiting values of special functions at roots of unity -perhaps closed formulas can be computed for d S beyond the cases of S being integers in arithmetic progressions, kth power-free integers, and other subsets of N with arithmetic densities already explicitly known.

n

  log log n < ∞, then for suitable subsets S of the positive integers Wang proves that -limN →∞ 2≤n≤N p min (n)∈S (µ * a)(n) n = d S , (1.3) where (f * g)(n) = d|n f (d)g(n/d) is the classical Dirichlet convolution of f and g. The first author thanks the support of the Thomas Jefferson Fund and the NSF (DMS-1601306 and DMS-2002265). We thank episciences.org for providing open access hosting of the electronic journal Hardy-Ramanujan Journal 1. Introduction and statement of results 2 1. Introduction and statement of results

  of classical identities, like a partition Cauchy product formula for absolutely convergent series λ∈P a(λ) λ∈P b(λ) = λ∈P δ|λ a(δ)b(λ/δ),(2.12)

4.

  Proof of Theorem 1.3 and its corollaries 10 4. Proof of Theorem 1.3 and its corollaries

.

  Swap a, b in Lemma 4.1, noting a * b = b * a, and let b(λ) = µ P (λ) to yield sm(λ)=n

  Proof of Corollary 1.6. With a, f, g and b = b a,f,g as defined in Corollary 1.6, following Wang's treatment, apply the partition Möbius inversion formula (2.11) to the definition of b a,f,g (λ) in the statement of the corollary to give (µ P * a) (λ) g (sm(λ)) f (sm(λ)) = δ|λ µ P (δ)b a,f,g (λ/δ) = (µ P * b)(λ). (4.23)

Table 2 :

 2 Illustration of d S (2)

	5

K. Ono, R. Schneider and I. Wagner, Partition-theoretic formulas for arithmetic densities, II

Conceptual background

Conceptual background

We note that the partition convolution operation was initially suggested to the second author by O. Beckwith (personal communication, 2018).

For q ∈ [0, 1), equation (3.14) can be interpreted as the expected value of the indicator function for the subset S, given the probability function P (n) = q n-1 (1 -q) for random n ∈ N.

Indeed, (3.15) holds if q → 1 in a "Stolz sector" of the unit circle; for a somewhat more general context, the reader is referred to the second author's study[Sc], following up on the present work.

The values in Table2were computed by A. V. Sills in Mathematica.

Computing arithmetic densities with q-series

Computing arithmetic densities with q-series

E.g. see Section 5. below.

Or any subset of N if q → 1 radially, by the result of Frobenius cited above.

A slightly more general theorem follows from using arbitrary a(∅) in this proof.

Further examples

Acknowledgement. We wish to express our gratitude to A.V. Sills for computing the values in Table 2, and to Paul Pollack for providing advice and references on Tauberian theorems. Moreover, the second author thanks G.E. Andrews and Prof. Sills for discussions on convergence, divergence and q-analogues that strongly informed this work.

Proof. For the first identity, set a(λ) = (λ) and f equal to the indicator function of S in Corollary 4.3. Then since ((n)) = 1, (∅) = 0, and (γ • (n)) -(γ) = 1 for any γ ∈ P, we have

(5.28)

Note from the right-hand side that A n (q) does not satisfy either of the analytic conditions of Theorem 1.3 as q → 1. However, by (5.28), it fits nicely into the right-hand side of Lemma 4.3:

A n (q)q n (q; q) n = n∈S q n (q; q) n-1 (q; q) n = n∈S q n 1 -q n , which reduces the corollary to a rewriting of Example 5.2 above. Dividing through by n≥1 q n 1-q n = n≥1 σ 0 (n)q n and letting q → 1 completes the proof of the first identity. We prove the second identity along very similar lines, setting a(λ) = sz(λ) = |λ|. Then since |(n)| = n, |∅| = 0, and |γ • (n)| -|γ| = n for any γ ∈ P, in this case we compute

(5.29)

Again, this A n (q) does not satisfy the conditions of Theorem 1.3; however, by (5.29), we have sm(λ)∈S (µ P * sz)(λ)q |λ| = (q; q) ∞ n∈S A n (q)q n (q; q) n = n∈S nq n (q; q) n-1 (q; q) n = n∈S nq n 1 -q n . Now, the right-hand side of these equalities can be rewritten

(5.30)

which, combined with (5.30) above, gives

using well-known Lambert series identities in the right-hand equalities. Dividing through by n≥1 σ 1 (n)q n and letting q → 1 gives the second identity.

Remark 5.8. Note that (λ) and sz(λ) are the r = 0 and r = 1 cases, respectively, of sz r (λ) :=

where the sum is over the parts λ i ∈ N of partition λ. 9 More generally, inserting sz r (λ) for any integer r ≥ 0 in place of (λ) and sz(λ), the steps of the proof of Example 5.7 continue to hold, but yielding (µ P * sz r )(λ) in the numerator and σ r (n) := d|n d r in the denominator.