ON THE TORUS

(1)

ANNALES DE

L’INSTITUT FOURIER

LesAnnales de l’institut Fouriersont membres du

Reda Chhaibi, Freddy Delbaen,

Pierre-Loïc Méliot & Ashkan Nikeghbali

Mod-φconvergence: Approximation of discrete measures and harmonic analysis on the torus

Tome 70, n^o3 (2020), p. 1115-1197.

<http://aif.centre-mersenne.org/item/AIF_2020__70_3_1115_0>

Cet article est mis à disposition selon les termes de la licence Creative Commons attribution – pas de modification 3.0 France.

http://creativecommons.org/licenses/by-nd/3.0/fr/

(2)

MOD-φ CONVERGENCE: APPROXIMATION OF DISCRETE MEASURES AND HARMONIC ANALYSIS

ON THE TORUS

by Reda CHHAIBI, Freddy DELBAEN, Pierre-Loïc MÉLIOT & Ashkan NIKEGHBALI

Abstract. —In this paper, we relate the framework of mod-φconvergence to the construction of approximation schemes for lattice-distributed random variables.

The point of view taken here is the one of Fourier analysis in the Wiener algebra, allowing the computation of asymptotic equivalents of the local, Kolmogorov and total variation distances. By using signed measures instead of probability measures, we are able to construct better approximations of discrete lattice distributions than the standard Poisson approximation. This theory applies to various examples arising from combinatorics and number theory: number of cycles in permutations, number of prime divisors of a random integer, number of irreducible factors of a random polynomial, etc. Our approach allows us to deal with approximations in higher dimensions as well. In this setting, we bring out the influence of the correlations between the components of the random vectors in our asymptotic formulas.

Résumé. —Dans cet article, nous relions la théorie de la convergence mod- φ à la construction de schémas d’approximation pour des variables aléatoires à valeurs dans des réseaux. Le point de vue adopté est celui de l’analyse de Fou- rier dans l’algèbre de Wiener ; il permet le calcul d’équivalents asymptotiques des distances locales, de Kolmogorov et en variation totale. En utilisant des mesures signées au lieu de mesures de probabilités, nous construisons des approximations de distributions discrètes meilleures que l’approximation standard poissonnienne.

Cette théorie s’applique à divers exemples issus de la combinatoire et de la théo- rie des nombres : nombre de cycles dans des permutations, nombre de diviseurs premiers d’un entier aléatoire, nombre de facteurs irréductibles d’un polynôme aléatoire, etc. Notre approche permet également des approximations en dimension supérieure. Dans ce cadre, nous mettons en évidence l’influence sur nos formules asymptotiques des corrélations entre les composantes des vecteurs aléatoires.

Keywords:Mod-φconvergence, Wiener algebra, Lattice distributions, Approximation of random variables.

2020Mathematics Subject Classification:62E17, 62E20, 60E10, 11N37, 11N45.

(3)

1. Introduction

1.1. Poisson approximation of lattice-valued random variables

Consider a sequence (Bi)i>1of independent Bernoulli random variables, withP[B_i= 1] =p_i andP[B_i= 0] = 1−p_i. We set

Xn=

n

X

i=1

Bi.

ThePoisson approximationensures that if thepi’s are small but their sum λ_n =Pn

i=1p_iis large, then the distribution ofX_nis close to the distribution of a Poisson random variable with parameterλn =Pn

i=1pi. To make this result quantitative, one can introduce the total variation distance between two probability measuresµandν onZ. Let us set

d_TV(µ, ν) =X

k∈Z

|µ({k})−ν({k})|;

this definition differs by a multiplicative factor 2 from the usual conven- tions, but otherwise we would have to keep track of a factor ¹₂ in all our estimates. A first quantitative result regarding the Poisson approximation is due to Prohorov and Kerstan (see [40, 49]): ifpi= ^λ_n for alli∈[[1, n]], then

dTV(Xn,P(λ)) =X

k∈N

P[Xn=k]−e^−λλ^k k!

62λ

n.

More generally, with parameters pi that can be distinct and with λn = Pn

i=1pi, Le Cam showed that X

k∈N

P[Xn =k]−e^−λⁿ (λ_n)^k k!

62

n

X

i=1

(pi)².

This is an immediate consequence of the inequality on total variation distances

d_TV(µ₁∗µ₂, ν₁∗ν₂)6d_TV(µ₁, ν₁) +d_TV(µ₂, ν₂)

which holds for any probability measuresµ1, µ2, ν1, ν2 on Z (cf. [14]). By using arguments derived from Stein’s method for the Gaussian approximation, Chen and later Barbour and Eagleson improved versions of this inequality, e.g.,

X

k∈N

P[X_n =k]−e^−λⁿ (λ_n)^k k!

62 (1−e⁻Pn i=1p_i

) Pn

i=1(pi)² Pn

i=1pi

,

(4)

see [5, 19, 54]. We also refer to [20] for an extension to possibly depen- dent Bernoulli random variables, to [2] for results regarding the Poisson processes, and to [6] for a survey of the theory of Poisson approximations.

More generally, consider random variablesXn>1whose distributions are supported by the latticeZ^d⊂R^d, and which stem from a common probabilistic model. In many cases, the scaling properties of the model imply that whennis large,Xn can be approximated by a discrete infinitely divisible lawν_n, the Lévy exponents of these reference laws being all proportional:

cνn(ξ) := X

k∈Z^d

νn(k) e^ihξ^|^ki= e^λⁿ^φ(ξ), λn →+∞.

To go beyond the classical Poisson approximation, one can try in this setting to write bounds on the total variation distancedTV(Xn, νn), or on another metric which measures the convergence in law (the local distance, the Kolmogorov distance, the Wasserstein metric, etc.). A convenient framework for this program is the notion ofmod-φconvergencedeveloped by Bar- bour, Kowalski and Nikeghbali in [7], see also the papers [23, 26, 27, 35, 41].

The main idea is that, given a sequence of random variables (Xn)_n∈Nwith values in a latticeZ^d, and a reference infinitely divisible lawφon the same lattice, if one has sufficiently good estimates on the Fourier transform of the lawµ_n ofX_n and on the ratio

bµ_n(ξ)

νbn(ξ) =E[e^iξXⁿ] e^−λⁿ^φ(ξ),

then one can deduce from these estimates the asymptotics of the distribution ofX_n: central limit Theorems [7, 35], local limit Theorems [23, 41], large deviations [26], speed of convergence [27], etc. In this paper, we shall use the framework of mod-φ convergence to compute the precise asymptotics of the distance between the law ofX_n and the reference infinitely divisible law, for various distances.

When approximating lattice-distributed random variables, another ob- jective that one can pursue consists in finding better approximations than the one given by an infinitely divisible law. A general principle is that, if one allows signed measures instead of positive probability measures, then simple deformations of the infinitely divisible reference law can be used to get smaller distances, i.e. faster convergences. In the setting of the classical Poisson approximation of sums of independent integer-valued random variables, this idea was used in [4, Theorem 5.1]. We also refer to [8, 13, 16, 17, 18, 36, 42, 48] for other applications of the signed compound Poisson approximation (SCP). For mod-Poisson convergent random variables (Xn)n∈N, an unconditional upper bound on the distance between the

(5)

law ofX_n and a signed measureν_n defined by means of Poisson–Charlier polynomials was proven in [7, Theorem 3.1]. A more general upper bound with an arbitrary reference law also appears in [7, Theorem 5.1]. In our paper, we shall set up a general approximation scheme for sequences of lattice-valued random variables, which will allow us:

• to obtain signed measure approximations whose distances to the laws of the variablesXn are arbitrary negative powers of the pa- rameterλ_n;

• to compute the asymptotics of these distances (instead of an unconditional upper bound).

Thus, this paper can be regarded as a complement to [7], with an alternative approach and an alternative goal (namely, obtaining the exact asymptotics of the distances). In [26, Proposition 4.1.1], we also used signed measures in order to approximate mod-φconvergent random variables, but with respect to a continuous infinitely divisible distribution φ, and using a first-order deformation of the Gaussian distribution.

In the remainder of this introductory section, we recall the main defini- tions from the theory of mod-φ convergence, and we explain the general approximation scheme. In subsequent sections, this approximation scheme will yield the main hypotheses of our Theorems (Sections 2 and 3), and we shall apply it to various one-dimensional and multi-dimensional examples coming from probability, analytic number theory, combinatorics, etc. (Sec- tions 4 and 5). We also present in this introduction the main tool that we shall use in this paper, namely, harmonic analysis in the Wiener algebra.

1.2. Infinitely divisible distributions and distances between probability measures

Let us start by presenting the reference infinitely divisible laws and the distances between probability distributions that we shall work with. Fix a dimensiond>1. IfX is a random variable with values inZ^d, we denote µ_X its probability law

µX(k1, k2, . . . , kd) =P[X= (k1, k2, . . . , kd)],

andµbXits Fourier transform, which is defined on the torusT^d= (R/2πZ)^d:

µb_X(ξ) =E[e^ihξ^|^Xi] = X

k1,...,kd∈Z

µ_X(k₁, . . . , k_d) exp



i

d

X

j=1

k_jξ_j



.

(6)

Assume that the law ofXis infinitely divisible. Then the Fourier transform ofX can be written uniquely as

µb_X(ξ) = e^φ(ξ), whereφis a function that is (2πZ)^d-periodic:

∀(n₁, . . . , n_d)∈Z^d, φ(ξ₁+ 2πn₁, . . . , ξ_d+ 2πn_d) =φ(ξ₁, . . . , ξ_d).

Moreover, the lawµX is supported onZ^d and none of its sublattices if and only if the Lévy–Khintchine exponentφis periodic with respect to (2πZ)^d and none of its sublattices. We refer to [53] and [55, Chapter 2] for details on lattice-distributed infinitely divisible laws (in the cased= 1); see also the discussion of [26, Section 3.1]. Throughout this paper, we make the following assumptions on a reference infinitely divisible lawµX:

• µX has moments of order 2;

• andµX is not supported on any sublattice ofZ^d.

Then, the corresponding Lévy–Khintchine exponentφis twice continuously differentiable and one has the Taylor expansion around zero

φ(ξ) = ihm|ξi −ξ^tΣξ

2 +o(|ξ|²),

where m = E[X], ξ^t is the transpose of the column vector ξ, and Σ is the covariance matrix of X, which is non-degenerate. Moreover, Re(φ(ξ)) admits a unique non-degenerate global maximum on [0,2π]^d atξ= 0.

Remark 1.1. — Let (γ ∈ R^d, A ∈ M(d×d,R),Π) be the triplet of the Lévy–Khintchine representation of the Fourier transform of an infinitely divisible random variableX (cf. [53, Theorem 8.1]). Then,X is supported onZ^d and has a moment of order 2 if and only if:

(1) A= 0 andγ∈Z^d;

(2) the Lévy measure Π is supported onZ^dand has a second moment.

Example 1.2. — Withd= 1, suppose thatX follows a Poisson distribu- tionP_(λ) with parameterλ. In this case,

φ(ξ) =λ e^iξ−1

= iλξ−λξ²

2 +o(ξ²).

Example 1.3. — More generally, fix a random variable Z with values in Z, and consider the compound Poisson distributionX =P^P(λ)

i=1 Zi,where theZ_i’s are independent copies ofZ, andP_(λ) is an independent Poisson variable. One obtains a new infinitely divisible law with values on the lattice

(7)

Z, and the corresponding Lévy–Khintchine exponent is φ(ξ) =λ E[e^iξZ]−1

= iλE[Z]ξ−λE[Z²]ξ²

2 +o(ξ²).

Example 1.4. — The previous example is generic if one restricts oneself to infinitely divisible random variables with values in N={0,1,2,3, . . .};

see [38, 52]. Thus, a random variableX with values inNis infinitely divisible if and only it admits a representation

X =

P_(λ)

X

i=1

Zi

as a compound Poisson distribution. In this representation, the Zi’s are independent copies of a random variableZ with law

P[Z =j] =λ_j

λ for allj >0;

andP(λ)is an independent Poisson variable with parameterλ=P∞ j=1λ_j. Then, another representation ofX is X =P∞

j=1j U_j, where the U_j’s are independent Poisson variables with parameters λj. These parameters λj

are the solutions of the system of equations kP[X =k] =

k

X

j=1

λ_jjP[X =k−j],

and this system provides a numerical criterion of infinite divisibility:X is infinitely divisible if and only if the solutionsλj are all non-negative.

Given a random variableX onZ^d, the general question that we want to tackle is: how close is the law µ=µX of X to an infinitely divisible law ν with exponent φ? For that purpose, one can choose different distances between probability measures, the typical ones being:

• thelocal distance:

dL(µ, ν) := sup

k∈Z^d

|µ({k})−ν({k})|.

• thetotal variation distance:

d_TV(µ, ν) := 2 sup

A⊂Z^d

|µ(A)−ν(A)|= X

k∈Z^d

|µ({k})−ν({k})|.

• and in dimension 1, theKolmogorov distance:

dK(µ, ν) := sup

k∈Z

|µ([[−∞, k]])−ν([[−∞, k]])|

= sup

k∈Z

|µ([[k,+∞]])−ν([[k,+∞]])|,

(8)

where [[a, b]] ={a, a+ 1, a+ 2, . . . , b}denotes an integer interval.

It is well known and immediate to check that

d_L(µ, ν)62d_K(µ, ν)6d_TV(µ, ν)

for any pair of signed measures (µ, ν). The hardest distance to estimate is usually the total variation distance. Note that all the previous quantities metrize the convergence of signed measures on Z^d with respect to the weak or strong topology. As recalled before in the Poisson case, many approaches for the estimation ofd_L, d_K and d_TV can be found in the lit- erature, the most popular ones being Stein’s method, coupling methods and semi-group methods, see [22, 54]. The use of characteristic functions has long been considered not so effective, until the remarkable series of papers of Hwang [32, 33, 34] (see also [30, Section IX.2]). In [34], Hwang explained how an effective Poisson approximation can be achieved assum- ing the analyticity of characteristic functions. In a similar spirit, but with much weaker hypotheses, [7] explored this question by using the notion of mod-φconvergence. Our paper revisits this notion, with the purpose of showing that the phenomenons at stake find their source in the harmonic analysis of the torus. While [7] was devoted to the proof ofunconditional upper bounds on the distancesdL,dK,dTV, here we shall be interested in theasymptotics of these distances.

1.3. Mod-φ convergent sequences of random variables We consider an infinitely divisible law ν of exponent φ on Z^d, and a sequence of random variables (X_n)_n∈_NonZ^d. Following [23, 26, 35], we say that (Xn)_n∈N converges mod-φ with parameters λn → +∞ and limiting functionψif

E h

e^ihξ^|^Xⁿⁱi

= e^λⁿ^φ(ξ)ψn(ξ) with

n→∞lim ψ_n(ξ) =ψ(ξ).

The precise hypotheses on the convergenceψ_n →ψ will be made explicit when needed; typically, we shall assume it to occur in some spaceC^r(T^d) endowed with the norm

kfk_Cr = sup

|α|6r

sup

ξ∈T^d

|(∂^αf)(ξ)|.

Let us provide a few examples which should enhance the understanding of this notion, as well as show its large scope of applications.

(9)

Example 1.5. — Consider as in Section 1.1 a sum of independent random variables

Xn=

n

X

i=1

Bi,

withBifollowing a lawB(pi):P[Bi= 1] = 1−P[Bi= 0] =pi. Let us assume that P∞

i=1pi = +∞ and P∞

i=1(pi)² < +∞. Then, settingλn = Pn i=1pi

andµn=µXn, µbn(ξ) =

n

Y

i=1

1 +pi(e^iξ−1)

= e^λⁿ^(e^iξ⁻¹⁾

n

Y

i=1

1 +pi(e^iξ−1)

e^−pⁱ^(e^iξ⁻¹⁾

= e^λⁿ^(e^iξ⁻¹⁾

n

Y

i=1

1−(pi(e^iξ−1))²

2 (1 +o(1))

= e^λⁿ^(e^iξ⁻¹⁾ψn(ξ) withψ_n(ξ)→ψ(ξ) =Q∞

i=1 1 +p_i(e^iξ−1)

e^−pⁱ^(e^iξ⁻¹⁾. This infinite product converges uniformly on the circle because of the hypothesisP∞

i=1(pi)²<

∞. So, one has mod-Poisson convergence with parametersλ_n.

The following two examples will later be generalized to the multidimen- sional setting.

Example 1.6. — Ifσis a permutation of the integers in [[1, n]], denote`(σ) its number of disjoint cycles, including the fixed points. We then set`_n =

`(σn), whereσn is taken at random uniformly among then! permutations of Sn. Using Feller’s coupling (cf. [1]), one can show that `n admits the following representation in law:

`n =

n

X

i=1

B 1

i

,

where the Bernoulli random variables are independent. This representation will also be made clear by the discussion of Section 5.3 in the present paper.

By the discussion of the previous example, (`_n)_n∈_Nconverges mod-Poisson with parametersHn =Pn

i=1 1

i and limiting function

∞

Y

i=1

1 +e^iξ−1 i

e⁻^ei

ξ−1

i = 1

Γ(e^iξ)e^γ^(e^iξ⁻¹⁾,

whereγis the Euler–Mascheroni constantγ= lim_n→∞(Hn−logn) (see [3]

for this infinite product representation of the Γ-function, due to Weier- strass). Thus, one can also say that (`n)_n∈N converges mod-Poisson with parametersλ_n= lognand limiting function

ψ(ξ) = 1 Γ(e^iξ).

(10)

Example 1.7. — As pointed out in [35, 41], mod-φconvergence is a common phenomenon in probabilistic number theory. For instance, denoteω(k) the number of distinct prime divisors of an integerk >1, and ω(N_n) the number of distinct prime divisors of a random integerNn smaller thann, taken according to the uniform law. Then, it can be shown by using the Selberg–Delange method (cf. [57, Section II.5]) that

E[e^zω(Nⁿ⁾] = 1 n

n

X

k=1

e^zω(k)= e^{(log log}^n)(e^z⁻¹⁾

Ψ(e^z) +O 1

logn

with

Ψ(z) = 1 Γ(z)

Y

pprime

1 + z−1 p

e⁻^z−1^p ,

and where the remainderO(_log¹_n) is uniform forzin a compact subset ofC. Therefore, one has mod-Poisson convergence at speedλ_n = log logn, and with limiting functionψ(ξ) = Ψ(e^iξ). This limiting function involves two factors: the limiting function _Γ(e¹_iξ₎ of the number of cycles of a random permutation (“geometric” factor), and an additional “arithmetic” factor Q

p∈P(1 +êîξ_p⁻¹) exp(−êîξ_p⁻¹). This appearance of two limiting factors is a common phenomenon in number theory [35, 41].

Let (X_n)_n∈_N be a sequence of random variables that converges mod- φ with parametersλn. Informally, ψn(ξ) = E[e^ihξ^|^Xⁿⁱ] e^−λⁿ^φ(ξ) measures a deconvolution residue, and mod-φ convergence means that this residue stabilizes, allowing the computation of equivalents ofd(µ_n, ν_n), where

µn= law ofXn;

νn= law of the infinitely divisible law with exponentλnφ

anddis one of the distances introduced in the previous paragraph. This setting will be called thebasic approximation scheme for a mod-φconvergent sequence.

1.4. The Wiener algebra and the general scheme of approximation

As explained before, we shall be interested in more general approximation schemes, with signed measures ν_n that are closer to µ_n than the basic scheme. When d = 1, the appearance of signed measures is natural in the setting of the Wiener algebra of absolutely convergent Fourier series (see [37, 39]):

(11)

Definition 1.8. — The Wiener algebra A = A(T) is the algebra of continuous functions on the circle whose Fourier series converges absolutely.

It is a Banach algebra for the pointwise product and the norm kfk_A :=X

n∈Z

|cn(f)|, wherecn(f)denotes the Fourier coefficientR2π

0 f(e^iθ) e^−inθ_2π^dθ.

Note that the characteristic function of any signed measure µ onZbe- longs to the Wiener algebra, with

kµkb _A =X

n∈Z

|cn(µ)|b =X

n∈Z

|µ(n)|=kµkTV

equal to the total variation norm of the measure. Thus, A is the right functional space in order to use harmonic analysis tools when dealing with (signed) measures. To the best of our knowledge, this interpretation of the total variation distance as the norm ofA has not been used before. On the other hand, another important property of the Wiener algebra is:

Proposition 1.9 (Wiener’s _f¹ theorem). — Let f ∈A. Thenf never vanishes on the circle if and only if ¹_f ∈A.

We refer to [45] for a short proof of the Wiener theorem. As a consequence, ifXnis aZ-valued random variable and ifνis an infinitely divisible distribution with exponentφ, then the deconvolution residue

ψn(ξ) =E[e^iξXⁿ] e^−λⁿ^φ(ξ)

belongs toA, so it is a convergent Fourier seriesψn(ξ) =P∞

k=−∞ak,ne^ikξ. The idea is then to replace this residueψ_n by a simpler residueχ_n ∈A, which after reconvolution by e^λⁿ^φ(ξ)yields a signed measure approximating the law ofXn. We are thus led to:

Definition 1.10. — Let (X_n)_n∈_N be a sequence of random variables in Z^d that is mod-φ convergent with parameters (λn)n∈N. A general approximation schemefor(X_n)_n∈_N is given by a sequence of discrete signed measures(νn)n∈N onZ^d, such that

µcn(ξ) =E[e^ihξ^|^Xⁿⁱ] = e^λⁿ^φ(ξ)ψn(ξ);

cνn(ξ) = e^λⁿ^φ(ξ)χn(ξ),

withlim_n→+∞ψn(ξ) =ψ(ξ)andlim_n→+∞χn(ξ) =χ(ξ). Here, the residues ψ_n, ψ, χ_n and χare functions on the torus T^d = (R/2πZ)^d that have absolutely convergent Fourier series, and with χn(0) = χ(0) = 1 (hence,

(12)

ν_n(Z^d) = 1). The convergence of the residuesψ_n →ψ andχ_n →χ is assumed to be at least uniform on the torus, that is in the space of continuous functionsC⁰(T).

The basic approximation scheme is the case when χ_n(ξ) = χ(ξ) = 1.

The residues χn and χ will typically be trigonometric polynomials, and they will enable us to enhance considerably the quality of our discrete approximations.

Example 1.11. — An important case of approximation scheme in the sense of Definition 1.10 is theapproximation scheme of orderr>1. Suppose thatd= 1 and that the functionsψ_n(ξ) can be represented on the torus as absolutely convergent series

ψn(ξ) = 1 +

∞

X

k=1

bk,n(e^iξ−1)^k+

n

X

k=1

ck,n(e^−iξ−1)^k. The coefficientsa_k,n ofψ_n(ξ) =P∞

k=−∞a_k,ne^ikξ are related to the coeffi- cientsbk,nandck,nby the equations

bk,n=X

l>k

l k

al,n ; ck,n=X

l>k

l k

a_−l,n

for allk>1. Set then χ^(r)_n (ξ) =P_n^(r)(e^iξ) = 1 +

r

X

k=1

b_k,n(e^iξ−1)^k+

r

X

k=1

c_k,n(e^−iξ−1)^k. Theχ^(r)n (ξ) are the Laurent polynomials of degreerthat approximate the residues ψn around 0 at order r > 1. Then, if Xn is a random variable with law µn with characteristic function µcn(ξ) = e^λⁿ^φ(ξ)ψn(ξ), the approximation scheme of orderrofµ_n is given by the signed measuresνn^(r), with

νdn^(r)(ξ) = e^λⁿ^φ(ξ)χ^(r)_n (ξ).

We shall prove in Section 3 thatd_L(µn, νn^(r)),d_K(µn, νn^(r)) andd_TV(µn, νn^(r)) get smaller whenrincreases; in particular,νn^(r^>¹⁾is asymptotically a better approximation ofµn than the basic scheme νn⁽⁰⁾.

One can give a functional interpretation to the approximation schemes of orderr>1. Denote as beforePn^(r)the Laurent approximation of orderrof ψ_n, and introduce the shift operator S on functionsf :Z→C, defined by

(Sf)(k) =f(k+ 1).

(13)

Ifνn^(r) is the signed measure with Fourier transform given by νdn^(r)(ξ) = e^λⁿ^φ(ξ)P_n^(r)(e^iξ),

then for any square-integrable functionf :Z→C, iffb(ξ) =P

k∈Zf(k) e^ikξ, then

ν_n^(r)(f) =X

k∈Z

ν_n^(r)({k})f(k) = 1 2π

Z 2π 0

νdn^(r)(ξ)fb(ξ) dξ

= 1 2π

Z 2π 0

νdn⁽⁰⁾(ξ)P_n^(r)(e^iξ)fb(ξ) dξ

= 1 2π

Z 2π 0

νdn⁽⁰⁾(ξ)(Pn\^(r)(S)f)(ξ) dξ

=ν_n⁽⁰⁾(P_n^(r)(S)f).

Moreover, the operatorPn^(r)(S) is a linear combination of discrete difference operators:

P_n^(r)(S) = id +

r

X

k=1

bk,n(∆+)^k+

r

X

k=1

ck,n(∆₋)^k with

(∆^k₊(f))(j) =

k

X

l=0

(−1)^k−l k

l

f(j+l);

(∆^k₋(f))(j) =

k

X

l=0

(−1)^k−l k

l

f(j−l).

Therefore:

Proposition 1.12. — In the previous setting, if (νn^(r))n∈N is the approximation scheme of order r > 1 of a sequence of probability measures(µ_n)n∈Nthat is mod-φconvergent, then for any square-integrable functionf,

ν_n^(r)(f) =E[(P_n^(r)(S)(f))(Y_n)],

where Y_n follows an infinitely divisible law with exponent λ_nφ. Thus, to estimate at orderrthe expectationµn(f) =E[f(Xn)]:

(1) one replacesX_n by an infinitely divisible random variableY_n; (2) and one also replaces the functionf byPn^(r)(S)(f), which is better

suited for discrete approximations.

(14)

Remark 1.13. — Our notion of approximation scheme should be com- pared to the one of [34], which is the particular case where the reference infinitely divisible law is Poissonian, and the residuesχ_nare constant equal to one. One of the main interest of our approach is that we are able to construct better schemes of approximation, by allowing quite general residues χn in the Fourier transform of the laws νn that approximate the random variables Xn. On the other hand, we consider only three distances among those studied in [34], but there should be no difficulty in adapting our results to the other distances, such as the Wasserstein metric and the Hellinger/Matusita metric.

Remark 1.14. — The approximation scheme of order r>1 can be considered as a discrete analogue of the Edgeworth expansion in the central limit theorem, which is with respect to the Gaussian approximation, and thus gives results for a different scale of fluctuations [47, Chapter VI].

1.5. Outline of the paper

Placing ourselves in the setting of a general approximation scheme, one basic idea in order to evaluate the distances between the distributionsµn

andν_n is to relate them to the distances between the two residues ψ_n(ξ) and χn(ξ) in the Wiener algebra A(T^d). In Section 2, we prove various concentration inequalities in this algebra, and we explain how to use them in order to compute distances between distributions. In Section 3, we apply these results to obtain asymptotic estimates for the distances in the general approximation scheme (see our main Theorems 3.3, 3.6 and 3.11). In these two sections, we shall restrict ourselves to the one-dimensional setting, postponing the more involved computations of the higher dimensionsd>2 to Section 5.

In Section 4, we apply our theorem to various one-dimensional examples of mod-Poisson convergent sequences:

• In Section 4.1, we explain how to use the general theory with the toy-model of sums of independent Bernoulli random variables (classical Poisson approximation). The combinatorics of the approximation schemes of orderr>1 can be encoded in the algebra of symmetric functions, and we explain this encoding in Section 4.2, by using the theory of formal alphabets. The formal alphabets provide a simple description of the approximation schemes for all the examples hereafter, although none of them (except the toy-model) come from independent Bernoulli variables.

(15)

• In Section 4.3, we study the number of disjoint cycles in a model of random permutations which generalises the uniform and the Ewens measure (Example 1.6). The mod-Poisson convergence of this model was proven in [46], and we compute here the approximation schemes of this model.

• In Section 4.4, we show more generally how to usegenerating series with algebraico-logarithmic singularities to construct general approximation schemes of statistics of random combinatorial objects.

We study with this method the number of irreducible factors of a random monic polynomial over the finite fieldFq (counted with or without multiplicity), and the number of connected components of a random functional graph.

• In Section 4.5, we consider the number of distinct prime divisors of a random integer (Example 1.7). We explain how to use the form of the residueψ(ξ) of mod-Poisson convergence to construct explicit approximation schemes of the corresponding probability measures, using again the formalism of symmetric functions.

Our approach also allows us to measure the gain of the Poissonian approximation in comparison to the Gaussian approximation, when one has a sequence of random integers that behaves asymptotically like a Poisson random variable with a large parameterλn. In Section 5, we extend our results to the multi-dimensional setting. One of the advantages of the Fourier approach to approximation of probability measures is that it allows one to deal with higher dimensions with the exact same techniques, although the computations are more involved. There is however an important difference from the one-dimensional setting: the dependence between the coordinates of a mod-φconvergent sequence (X_n)n∈N in Z^d has an influence over the asymptotics of the distances between the laws of the random variables and their approximation schemes. Note that this happens even when the reference infinitely divisible distribution corresponds to independent coordinates. We provide two examples of this phenomenon, stemming respectively from the combinatorics of the wreath productsSno(Z/dZ) and from the number theory of residue classes of prime numbers.

2. Concentration inequalities in the Wiener algebra For convenience, until Section 5, we shall focus on the one-dimensional case (d= 1) and the corresponding torusT=R/2πZ, which we view as the

(16)

set of complex numbers of modulus 1. Thus, a function onTwill be a function of e^iξ withξ ∈[0,2π). For p∈[1,+∞), the space of complex-valued functions on T whose p-powers are Lebesgue integrable will be denoted L^p =L^p(T); it is a Banach space for the norm

kfk_p:=

Z 2π 0

|f(e^iξ)|^p dt 2π

¹p

.

Forp= +∞, L^∞(T) is the space of essentially bounded functions on the torus, with Banach space norm

kfk_∞:= ess-sup_ξ∈[0,2π]

f(e^iξ) .

In the sequel, we abbreviate sometimes the Haar integralR2π

0 f(e^iξ)^dξ_2π by R

Tf(e^iξ), or simplyR

Tf.

2.1. Deconvolution residues in the Wiener algebra

Recall that the Wiener algebraA(T) is the complex algebra of absolutely convergent Fourier series, endowed with the norm kfk_A =P

n∈Z|cn(f)|.

We shall use the following property ofA(T), which is akin to Poincaré’s inequality for Sobolev spaces (see [39, Section 6.2]):

Proposition 2.1. — There is a constant C_H = ^√^π

3 = 1.814. . . such that

kfk_A 6|c0(f)|+CHkf⁰k_L² for allf ∈A.

Proof. — Combining the Cauchy–Schwarz inequality and the Parseval identity, we obtain

kfk_A =|c0(f)|+X

n6=0

1

n|n cn(f)|

6|c0(f)|+ s

X

n6=0

1 n²

sX

n6=0

|n cn(f)|²

6|c0(f)|+ rπ²

3 kf⁰k_L2.

Note that the inequality is sharp, and it implies that the Sobolev space W^1,2(T) ofL²functions onTwith weak derivative inL² is topologically

included in the Wiener algebra.

(17)

Another important tool for the computation of total variation distances is the following Lemma 2.2. Calldeconvolution residueof a signed measure µonZby another signed measureν onZthe function

δ(ξ) :=µ(ξ) (b νb(ξ))⁻¹.

By Wiener’s theorem, if νb never vanishes, which is for instance the case when it is the Fourier transform of an infinitely divisible law ν [53, Lemma 7.5], thenbν⁻¹∈A and the previous deconvolution happens in the Wiener algebra: δ ∈ A. Now, in order to measure the distance between the lawµofX and a reference lawν, one can adopt the following point of view, which is common in signal processing. For the deconvolution residue to be considered as a small noise,δhas to be close to the constant function 1 (the Fourier transform of the Dirac distribution at zero). In particular, suppose that we are given a general scheme of approximation of a sequence of random variables (Xn)_n∈_Nby a sequence of laws (νn)_n∈_N. Then, with the notation of Definition 1.10, one has a sequence of deconvolution residues

δ_n(ξ) =µc_n(ξ) cνn(ξ),

and the quality of our scheme of approximation will be related to the speed of convergence of (δn)n∈Ntowards the constant function 1. More precisely:

Lemma 2.2 (Fundamental inequality for deconvolution residues). Letµandνbe two signed measures onZandδ= b^µ

bν be the deconvolution residue (we assume thatνbdoes not vanish on T). For anyc∈A,

dTV(µ, ν)6kc(δ−1)bνk_A +kδ−1k_Ak(1−c)bνk_A. Proof.

dTV(µ, ν) =kbµ−bνk_A =k(δ−1)νbk_A

6kc(δ−1)bνk_A +k(1−c)(δ−1)νkb _A

6kc(δ−1)bνk_A +kδ−1k_Ak(1−c)bνk_A. In practice, c ∈A will be a carefully chosen cut function concentrated on regions wherebν is large. So, informally,ν is a good approximation ofµ if the deconvolution residueδ:

• is close to 1 whereνbis large;

• has a reasonable norm in the Wiener algebra.

Note that the fundamental inequality does rely on the algebra norm of A(T). We now quantify this idea.

(18)

2.2. Estimates of norms in the Wiener algebra

In this paragraph, we fix two discrete measuresµandν onZ, both being convolutions of an infinitely divisible law with exponentλφby residuesψ andχ:

µ(ξ) = eb ^λφ(ξ)ψ(ξ);

ν(ξ) = eb ^λφ(ξ)χ(ξ).

In Section 3, we shall recover the setting of a general approximation scheme by adding indices n to this situation. We denote m and σ² the first two coefficients of the Taylor expansion ofφaround 0:

φ(ξ) =miξ−σ²ξ²

2 +o(ξ²).

We then fix an integerr>0 such that:

(1) the residuesψandχare assumed to be (r+ 1) times continuously differentiable on [−ε, ε] for a certain ε >0;

(2) their Taylor expansion coincides at 0 up to ther-th order:

∀s∈[[0, r]], (ψ−χ)^(s)(0) = 0.

In this setting, we define the non-negative quantities βr+1(ε) = sup

ξ∈[−ε,ε]

(ψ−χ)^(r+1)(ξ) ; γ(ε) = sup

ξ∈[−ε,ε]

φ⁰⁰(ξ) +σ² ; M =− sup

ξ∈[−π,π]

Re(φ(ξ)) ξ²

.

For instance, ifφ(ξ) = e^iξ−1 is the exponent of the standard Poisson law, thenγ(ε) = 2 sin(^ε₂) andM =_π²₂. The strict positivity ofM as soon asφis not the constant distribution concentrated at 0 is proven in [26, Section 3].

Note that for anyξ∈[−ε, ε], one has Re(φ(ξ)) =

Z ξ θ=0

(ξ−θ) Re(φ⁰⁰(θ)) dθ

=−σ²ξ²

2 +

Z ξ θ=0

(ξ−θ) Re(φ⁰⁰(θ) +σ²) dθ6(γ(ε)−σ²)ξ²

2 .

On the other hand, outside the interval [−ε, ε], one can use the inequality Re(φ(ξ))6−M ξ².

(19)

Theorem 2.3(Norm estimate). — Takeεsmall enough so thatγ(ε)6

σ²

2, and suppose thatλis large enough so thatλ> σ²². Then:

kbµ−bνk_A 6kψ−χk_A 1 +CH

r 2

πε+λkφ⁰k_∞

!!

e⁻^{λM ε}

2 4

+Cr+1βr+1(ε) (ε⁻¹+p

5(r+ 1)) (^σ₂²λ)^r²⁺¹⁴ , whereCr+1 is a constant depending only onr:

Cr+1= 1 (r+ 1)!

s 2π

3 Γ

r+3 2

.

In order to prove this theorem, we introduce the cut function ξ∈T7→

c(ξ) that is piecewise linear, vanishes outside of [−ε, ε], and is equal to 1 on [−^ε₂,^ε₂], see Figure 2.1.

−^ε₂ ₂^ε

ε ε

ξ c(ξ)

Figure 2.1. The cut-functionc(ξ).

Lemma 2.4. — Under the assumptions of Theorem 2.3, (1−c) e^λφ

_A 6 1 +CH

r 2

πε+λkφ⁰k_∞

!!

e⁻^{λM ε}

2 4 .

Proof. — Note that|(1−c(ξ)) e^λφ(ξ)|vanishes on [−^ε₂,^ε₂], and is smaller than e^{−λM ξ}² 6e⁻^{λM ε}⁴ ² everywhere else. We then use Proposition 2.1:

(1−c) e^λφ A 6

Z

T

|1−c|e^λ^Re(φ)+CH

−c⁰e^λφ+ (1−c)λφ⁰e^λφ L²

6e⁻^{λM ε}

2 4 +CH

c⁰e^λφ

_L₂+CHλkφ⁰k_∞

(1−c) e^λφ _L₂

(20)

6e⁻^{λM ε}

2

4 +C_Hkc⁰k_L²e⁻^{λM ε}

2

4 +C_Hλkφ⁰k∞e⁻^{λM ε}

2 4

6 1 +C_H r 2

πε+λkφ⁰k∞

!!

e⁻^{λM ε}

2

4 .

Lemma 2.5. — Under the assumptions of Theorem 2.3, we have:

c(ψ−χ) e^λφ

_A 6Cr+1βr+1(ε) (ε⁻¹+p

5(r+ 1)) (^σ₂²λ)^r²⁺¹⁴ .

Proof. — Note that the normk · k_A is invariant after multiplication by e^ikξ, k∈Z. In particular,

c(ψ−χ) e^λφ _A =

c(ξ) (ψ(ξ)−χ(ξ)) eλφ(ξ)−iξbλmc A,

whereb · cdenotes the integer part of a real number. Using again Proposi- tion 2.1, we obtain

c(ψ−χ) e^λφ _A 6

Z

T

|c(ψ−χ) e^λφ|+CH

c(ξ) (ψ(ξ)−χ(ξ)) eλφ(ξ)−iξbλmc0 _L₂ 6

Z

T

|c(ψ−χ) e^λφ|+C_H

c⁰(ψ−χ) e^λφ L²

+CH

c(ψ−χ)⁰e^λφ

_L₂+CH

c(ψ−χ) (λφ⁰−ibλmc) e^λφ _L₂ 6A+B+C+D.

Let us bound separately each term. Recall thatR∞

−∞|t|^re^−t²dt= Γ ^r+1₂ forr>0.

(A) Forξ∈[−ε, ε],

|ψ(ξ)−χ(ξ)|6 βr+1(ε)|ξ|^r+1 (r+ 1)! ; e^λ^Re(φ(ξ))6e⁻^λ(σ

2−γ(δ))ξ2

2 6e⁻^λσ

2ξ2 4

the last inequality following from the assumptions on ε. Conse- quently,

A6 βr+1(ε) 2π(r+ 1)!

Z ε

−ε

e⁻^λσ

2ξ2

4 |ξ|^r+1dt= βr+1(ε)

2π(r+ 1)! (λ^σ₄²)^r²⁺¹ Γr 2+ 1 6 β_r+1(ε)

2π(r+ 1)! (λ^σ₄²)^r²⁺¹ s

Γ r

2+3 4

Γ

r 2+5

4

(21)

6 βr+1(ε) (r+ 1)! (λ^σ₂²)^r²⁺¹

2 π

¹₄ s 1 2πΓ

r+3

2

.

by using the log-convexity of the Gamma function, and the dupli- cation formula Γ(2z) = ²^2z−1^√_π Γ(z) Γ(z+¹₂).

(B) Since|c⁰(ξ)|6²_εon [−ε, ε] and vanishes outside [−ε, ε], one obtains

B6 2CHβr+1(ε) ε(r+ 1)!

1 2π

Z ∞

−∞

e⁻^λσ

2ξ2

2 |ξ|^2r+2dξ ¹₂

6 β_r+1(ε) (r+ 1)! (λ^σ₂²)^r²⁺³⁴

2C_H ε

s 1 2πΓ

r+3

2

.

(C) Forξ∈[−ε, ε],

|(ψ−χ)⁰(ξ)|6βr+1(ε)|ξ|^r r! ,

so as before,

C6 C_Hβ_r+1(ε) r!

1 2π

Z ∞

−∞

e⁻^λσ

2ξ2

2 |ξ|^2rdξ ¹₂

6 CHβr+1(ε) r! (λ^σ₂²)^r²⁺¹⁴

s 1 2πΓ

r+1

2

6 βr+1(ε) (r+ 1)! (λ^σ₂²)^r²⁺¹⁴

CH(r+ 1) q

r+¹₂ s

1 2πΓ

r+3

2

.

(D) Last, forξ∈[−ε, ε],|φ⁰(ξ)−iµ|6γ(δ)|ξ|6 ^σ²2^|ξ|, and then,

|(λφ⁰(ξ)−ibλµc)|6 λσ²|ξ|

2 +|λµ− bλµc|6λσ²|ξ|

2 + 1.

Therefore,

D6 CHβr+1(ε)

√

2π(r+ 1)!

sZ

R

e⁻^λσ²²^ξ² |ξ|^2r+2dξ+λ σ² 2

sZ

R

e⁻^λσ²²^ξ² |ξ|^2r+4dξ

!

6 CHβr+1(ε)

√2π(r+ 1)!

1 (λ^σ₂²)^r²⁺³⁴

s Γ

r+3

2

+ 1

(λ^σ₂²)^r²⁺¹⁴ s

Γ

r+5 2

! .