Convergence in distribution norms in the CLT for non identical distributed random variables

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Convergence in distribution norms in the CLT for non identical distributed random variables

Vlad Bally, Lucia Caramellino, Guillaume Poly

To cite this version:

Vlad Bally, Lucia Caramellino, Guillaume Poly. Convergence in distribution norms in the CLT for non identical distributed random variables. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2018, 23, paper 45, 51 p. �10.1214/18-EJP174�. �hal-01413548�

arXiv:1606.01629v1 [math.PR] 6 Jun 2016

Convergence in distribution norms in the CLT for non identical distributed random variables

Vlad Bally^∗ Lucia Caramellino^†

Guillaume Poly^‡ June 7, 2016

Abstract

We study the convergence in distribution norms in the Central Limit Theorem for non identical distributed random variables that is

εⁿ(f) :=E

fXⁿ

i=1

Zⁱ

−E f(G)

→0

whereSⁿ=Pⁿ

i=1ZⁱwithZⁱ centred independent random variables (with a suitable re-normaliza- tion forSⁿ) andGis standard normal. We also consider local developments (Edgeworth expansion).

This kind of results is well understood in the case of smooth test functionsf. If one deals with measurable and bounded test functions (convergence in total variation distance), a well known theorem due to Prohorov shows that some regularity condition for the law of the random variables Xn,n∈N, on hand is needed. Essentially, one needs that the law ofXn is locally lower bounded by the Lebesgue measure (Doeblin’s condition). This topic is also widely discussed in the literature (see Battacharaya and Rao [12]). Our main contribution is to discuss convergence in distribution norms, that is to replace the test functionf by some derivative∂^αf and to obtain upper bounds forεn(∂^αf) in terms of the original functionf.

Contents

1 Introduction 2

2 Smooth test functions 5

2.1 Notation and main result . . . 5 2.2 Basic decomposition and proof of the main result . . . 7

3 General test functions 16

3.1 Differential calculus based on a splitting method . . . 16 3.2 CLT and Edgeworth’s development . . . 21

∗Universit´e Paris-Est, LAMA (UMR CNRS, UPEMLV, UPEC), MathRisk INRIA, F-77454 Marne-la-Vall´ee, France.

Email: bally@univ-mlv.fr

†Dipartimento di Matematica, Universit`a di Roma - Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma, Italy.

Email: caramell@mat.uniroma2.it. Corresponding author.

‡IRMAR, Universit´e de Rennes 1, 263 avenue du G´en´eral Leclerc, CS 74205 35042 Rennes, France.

4 Examples 32 4.1 An invariance principle related to the local time . . . 32 4.2 Convergence in distribution norms . . . 37

A Computation of the first three coefficients 39

B A Backward Taylor formula 41

References 42

1 Introduction

We consider a sequence of centred independent random variables Z_k ∈ R^d, k ∈ N with covariance matrixes σ_k^i,j =E(Z_kⁱZ_k^j) and we look to

S_n(Z) = Xn

k=1

Z_k. (1.1)

Our aim is to obtain a Central Limit Theorem as well as Edgeworth developments in this framework.

The basic hypotheses are the following. We assume the normalization condition Xn

k=1

σ_k =I_d (1.2)

where I_d ∈ Md×d is the identity matrix. Moreover we assume that for each p ∈ N there exists a constant Cp≥1 such that

maxk

E(|Z_k|^p)≤ C_p(Z)

n^p/2 . (1.3)

Letkfk_k,∞ denote the norm inW^k,∞,that is the uniform norm of f and of all its derivatives of order less or equal tok.First, we want to prove that

E(f(S_n(Z))− Z

Rd

f(x)γ_d(x)dx≤ C₀

n¹² kfk_3,∞ (1.4)

where γ_d(x) = (2π)^−d/2exp(−¹₂|x|²) is the density of the standard normal law. This corresponds to the Central Limit Theorem (hereafter CLT). Moreover we look for some functions (polynomials) ψ_k :R^d→R such that forN ∈Nand for every f ∈C_b^(N+1)(N+3)(R^d)

E(f(S_n(Z))− Z

Rd

f(x)X^N

k=0

1

n^k/2ψ_k(x)

γ_d(x)dx ≤ C_N

n^12(N+1)kfk_{(N+1)(N+3),∞}. (1.5) This is the Edgeworth development of order N. In the case of smooth test functions f (as it is the case in (1.5)), this topic has been widely discussed and well understood: such development has been obtained by Sirazhdinov and Mamatov [21] in the case of identically distributed random variables and then by G¨otze and Hipp [16] in the non identically distributed case. A complete presentation of this topic may be found in the book of Battacharaya and Rao [12]. It it worth to mention that the classical approach used in the above papers is based on Fourier analysis. In particular, the coefficients ψ_k in the above development are given as inverse Fourier transform of some suitable functions, so the expression ofψ_k is not completely transparent and its explicit computation requires some effort.

In our paper we use a different approach based on the Lindemberg method for Markov semigroups (this is inspired from works concerning the parametrix method for Markov semigroups in [9]). This alternative approach is convenient for the proof of our main result concerning “distribution norms”

(see below). But, even in the case of smooth test functions, this allows to obtain slightly more clear and precise results: we prove that ψ_k are linear combination of Hermite polynomials of order less or equal tok, whose coefficients are explicit and computed starting with the moments of Zi and Gi, Gi

denoting a Gaussian random variable with the same covariance matrix asZ_i.So the computation of these coefficients is easier. Moreover, our estimates hold for each fixedn(in contrast with the ones in the above papers, which are just asymptotic).

A second problem is to obtain the estimate (1.5) for test functionsfwhich are not regular, in particular to replace kfk(N+1)(N+3),∞ by kfk_∞.This amounts to estimate the error in total variation distance.

In the case of identically distributed random variables, and forN = 0 (so at the level of the standard CLT), this problem has been widely studied. First of all, one may prove the convergence in Kolmogorov distance, that is forf = 1D whereD is a rectangle. Many refinements of this type of result has been obtained by Battacharaya and Rao and they are presented in [12]. But it turns out that one may not prove such a result for a general measurable set D without assuming more regularity on the law of Z_k, k ∈ N. Indeed, in his seminal paper [20] Prohorov proved that the convergence in total variation distance is equivalent to the fact that there existsm such that the law ofZ₁+· · ·+Z_m has an absolutely continuous component. In [3] Bally and Caramellino obtained (1.5) in total variation distance, for identically distributed random variables, under the hypothesis that the law ofZ_kis locally lower bounded by the Lebesgue measure. We assume this type of hypothesis in this paper also. More precisely we assume that there existsr, ε >0 and there existsz_k∈R^d such that for every measurable setA⊂B_r(z_k)

P(Z_k∈A)≥ελ(A) (1.6)

where λ is the Lebesgue measure. This condition is known in the literature as Doeblin’s condition.

Under this hypothesis we are able to obtain (1.5) in total variation distance. It is clear that (1.6) is more restrictive than Prohorov’s condition. However we prove that in the framework of the CLT for identically distributed random variables, if we have Prohorov’s condition we may produce doubling condition as well, just working with the packagesY_k=P2(k+1)m

i=2km+1Z_i.This allows us to prove Corollary 3.12 which is a stronger version of Prohorov’s theorem.

Let us finally mention another line of research which has been strongly developed in the last years: it consists in estimating the convergence in the CLT in entropy distance. This starts with the papers of Barron [11] and Johnson and Barron [14]. In these papers the case of identically distributed random variables is considered, but recently, in [13] Bobkov, Chistyakov and G¨otze obtained the estimate in entropy distance for the case of random variables which are no more identically distributed as well.

We recall that the convergence in entropy distance implies the convergence in total variation distance, so such results are stronger. However, in order to work in entropy distance one has to assume that the law of Z_k is absolutely continuous with respect to the Lebesgue measure and have finite entropy and this is more limiting than (1.6). So the hypothesis and the results are slightly different.

A third problem is to obtain the CLT and the Edgeworth development with the test function f replaced by a derivative∂γf.If the law ofSn(Z) is absolutely continuous with respect to the Lebesgue measure, this means that we prove the convergence of the density and of its derivatives as well (which corresponds to the convergence in distribution norms). Unfortunately we fail to obtain such a result in the general framework: this is moral because we do not assume that the laws of Z_k, k = 1, ..., n are absolutely continuous, and then the law of S_n(Z) may have atoms. However we obtain a similar result, but we have to keep a “small error”. Let us give a precise statement of our result. For a functionf ∈C_p^m(R^d) (m times differentiable with polynomial growth) we defineL_m(f) and l_m(f) to

be two constants such that

X

0≤|α|≤m

|∂_αf(x)| ≤L_m(f)(1 +|x|)^lm(f).. (1.7) Our main result is the following: for a fixed m ∈ N, there exist some constants C_N ≥ 1 ≥ c_N > 0 (depending on r, ε from (1.6) and onC_p from (1.3)) such that for every multi-index γ with |γ|=m and for every f ∈C_p^m(R^d)

E

∂_γf(S_n(Z))− Z

Rd

∂_γf(x)X^N

q=0

1

n^q/2ψ_q(x)

γ_d(x)dx

≤CN

Lm(f)e^−cN×n+ 1

n^12(N+1)L0(f) .

(1.8)

If the random variables Z_k, k ∈ N are identically distributed we succeed to obtain exactly the same result under the Prohorov’s condition (see Corollary 3.12). So this is a strictly stronger version of Prohorov’s theorem (for m = 0 we get the convergence in total variation). Moreover, such result is used in [6] in order to give invariance principles concerning the variance of the number of zeros of trigonometric polynomials.

However we fail to get convergence in distribution norms becauseL_m(f)e^−cN×n appears in the upper bound of the error and L_m(f) depends on the derivatives of f. But we are close to such a result:

notice first that iffn=f∗φ_δn is a regularization by convolution with δn= exp(−^c_2m^N ×n) then (1.8) gives

E

∂_γf_n(S_n(Z))− Z

Rd

∂_γf_n(x)X^N

q=0

1

n^q/2ψ_q(x)

γ_d(x)dx≤ C_N

n^12(N+1)L₀(f). (1.9) Another way to eliminate L_m(f)e^−cN×n is to assume that the law of Z_i, i = 1, ..., m are absolutely continuous with the derivative of the density belonging to L¹. This is done in Proposition 4.2: we prove that for every k∈Nand every multi-indexα

sup

x

(1 +|x|²)^k|∂_αp_Sn(x)−∂_αγ(x)| ≤ C

√n

so, under these stronger conditions, we succeed to obtain convergence in distribution norms.

But the most interesting consequence of our result is given in Theorem 4.1: there we give an invariance principle for the occupation time of a random walk. More precisely we take ε_n = n^{−12(1−ρ)} with ρ∈(0,1) and we prove that, for everyρ^′ < ρ

Xn

k=1

E1

ε_n1_(−εn,εn)( Xk

i=1

Z_i)

−E Z ¹

0

1

ε_n1_(−εn,εn)(W_s)ds≤ C n^12(1+ρ′) withW_sa Brownian motion (so R1

0 1

εn1_(−εn,εn)(W_s)dsconverges to the local time ofW).Here the test function is fn= _ε¹_n1_(−εn,εn) and this converges to the Dirac function. This example shows that (1.8) is an appropriate estimate in order to deal with some singular problems.

The paper is organized as follows. In Section 2 we prove the result for smooth test functions (that is (1.5)) and in Section 3 we treat the case of measurable test functions. In order to do it we use some integration by parts technology which has already been used in [3] and which is presented in Section 3.1. We mention that a similar approach has been used by Nourdin and Poly [18], by using the Γ-calculus settled in [10]. The main result in Section 3 is Theorem 3.8. In Section 4 we treat

the two applications mentioned above. Finally we leave for Appendix A the explicit calculus of the coefficients ψ_q from (1.5) for q = 1,2,3 and in Appendix B we prove a technical result which is used in our development.

Although many ideas in our paper come from previous works (mainly from Malliavin calculus), at the end we finish with an approach which is fairly simple and elementary - so we try to give here a presen- tation which is essentially self contained (even if some cumbersome and straightforward computations are just sketched).

2 Smooth test functions

2.1 Notation and main result

We fix n ∈ N and we consider n centred and independent random variables Z = (Z_k)_1≤k≤n with Z_k= (Z_k¹, ..., Z_k^d)∈R^d. We denote by σ_k the covariance matrix of Z_k that is

σ_k^i,j =E(Z_kⁱZ_k^j), 1≤k≤n.

We look to

S_n(Z) = Xn

k=1

Z_k. (2.1)

Our aim is to compare the law of S_n(Z) with the law of S_n(G) where G = (G_k)_1≤k≤n denotes n centred and independent Gaussian random variables with the same covariance matrices:

E(Gⁱ_kG^j_k) =σ_k^i,j.

This is a CLT result (but we stress that it is not asymptotic). And we will obtain an Edgeworth development as well.

We assume thatZ_k has finite moments of any order and more precisely,

1max≤k≤n

E(|Z_k|ⁱ)≤ C_i(Z)

n^i/2 . (2.2)

In particular, fori= 2 the inequality (2.2) gives

1max≤k≤nsup

i,j |σ_k^i,j| ≤ C₂(Z)

n . (2.3)

Since the covariance matrix of G_k is equal to that of Z_k, the inequality (2.2) holds for the G_k’s as well, so we can resume by writing

sup

1≤k≤n

E(|Z_k|ⁱ)∨E(|G_k|ⁱ)≤ C_i(Z)

n^i/2 . (2.4)

Without loss of generality, (from H¨older) we can assume that 1 ≤ Ci(Z) ≤ Ci+1(Z) and more in general

1≤C_p(Z)≤C_pq^1/q(Z), p, q≥1.

Remark 2.1. Although it is not explicitly written, we are assuming that we fixn and that the laws of Z_k and G_k, as well as σ_k, are all depending on n. In our applications, we take a sequenceY ={Y_k}k

of i.i.d. centred r.v’s taking values inR^m and we consider Z_k = √¹

nC_kY_k, where C_k denotes ad×m matrix. Therefore, we actually study

S_n= 1

√n Xn

k=1

C_kY_k Notice that

1max≤k≤n

E(|Z_k|ⁱ)≤ ci(Y)

n^i/2 × max

1≤k≤nkC_kkⁱ,

in whichc_i(Y)denotes a constant depending only on (the law of ) theY_k’s, so that (2.4) actually holds.

We will specialize the results to this case. But in order to relax the notation and the proofs, it is much more useful to consider a general Z_k instead of √¹

nC_kY_k.

In order to give the expression of the terms which appear in the Edgeworth development we need to introduce some notation.

We say that α is a multiindex if α ∈ {1, . . . , d}^k for some k ≥ 1, and we set |α|= k its length. We allow the case k= 0, giving the void multiindex α=∅.

Letα be a multiindex and set k=|α|. For forx ∈R^d and f : R^d→ R, we denote x^α =xα1· · ·xα_k

and∂_αf(x) =∂_xα1· · ·∂_xαkf(x), the casek= 0 givingx^∅ = 1 and∂_∅f =f. In the following, we denote withC^k(R^d) the set of the functionsf such that∂_αf exists and is continuous for any αwith |α| ≤k.

The set C_p^k(R^d), resp. C_b^k(R^d), is the subset of C^k(R^d) such that ∂_αf has polynomial growth, resp.

is bounded, for anyα with|α| ≤k. C^∞(R^d), resp. C_p^∞(R^d) and C_b^∞(R^d), denotes the intersection of C^k(R^d), resp. of C_p^k(R^d) and ofC_b^k(R^d), for every k.

Forf ∈C_b^k(R^d) we denote

kfkk,∞=kfk_∞+ X

1≤|α|≤k

k∂_αfk_∞

and for f ∈C_p^k(R^d) we define L_k(f) andl_k(f) to be some constants such that X

0≤|α|≤k

|∂αf(x)| ≤L_k(f)(1 +|x|)^lk(f). (2.5) Moreover, for a non negative definite matrix σ ∈ Md×d we denote by L_σ the Laplace operator associated to σ, i.e.

Lσ = Xd

i,j=1

σ^i,j∂zi∂zj. (2.6)

Forr ≥1 andl≥0 we set

∆_α(r) =E(Z_r^α)−E(G^α_r) and D_r^(l) = X

|α|=l

∆_α(r)∂_α. (2.7)

Notice that D_r^(l) ≡0 for l= 0,1,2 and, by (2.4), for l≥3 and|α|=l then

|∆_α(r)| ≤ 2C_l(Z)

n^l/2 , r= 1, . . . , n. (2.8)

We construct now the coefficients of our development. Let N be fixed: this is the order of the development that we will obtain. Given 1≤m≤k≤N we define

Λm = {((l1, l^′₁), ...,(lm, l_m^′ )) :N+ 2≥li ≥3, N := [N/2]≥l^′_i≥0, i= 1, ..., m}, Λ_m,k = {((l₁, l^′₁), ...,(l_m, l_m^′ ))∈Λ_m :

Xm

i=1

l_i+ 2 Xm

i=1

l_i^′ =k+ 2m}. (2.9)

Then, for 1≤k≤N,we define the differential operator Γ_k =

Xk

m=1

X

((l1,l₁^′),...,(lm,l^′_m))∈Λ_m,k

X

1≤r1<...<rm≤n

Ym

i=1

1 l_i!D^(l_riⁱ⁾

Ym

j=1

(−1)^l′j 2^lj′l^′_j! L^l

j′

σ_rj. (2.10) By using (2.2) and (2.8), one easily gets the following estimates:

|Γ_kf(x)| ≤C× C_3k n^k/2 sup

|α|≤3k|∂_αf(x)|, f ∈C_b^3k(R^d), (2.11)

|Γ_kf(x)| ≤C× C_3k

n^k/2L_3k(f)(1 +|x|)^l3k(f), f ∈C_p^3k(R^d), (2.12) whereL_3k(f) andl_3k(f) are given in (2.5) and C,C_3k are positive constants.

We introduce now the Hermite polynomials. We refer to Nualart [19] for definitions and properties, here we just give the shortest way to introduce them by means of the integration by parts formula.

Given a multi-indexα, the Hermite polynomialH_α onR^d is defined by

E(∂_αf(W)) =E(f(W)H_α(W)) ∀f ∈C_p^∞(R^d) (2.13) where W is a standard normal random variable in R^d. Moreover for a differential operator Γ = P

|α|≤ka(α)∂_α,witha(α)∈R,we denote H_Γ=P

|α|≤ka(α)H_α so that

E(Γf(W)) =E(f(W)H_Γ(W)). (2.14)

Finally we define

Φ_N(x) = 1 + XN

k=1

H_Γk(x) with Γ_k defined in (2.10). (2.15) The main result in this section is the following (recall the constants L_k(f) and l_k(f), f ∈ C_p^k(R^d), defined in (2.5)):

Theorem 2.2. Let N ∈Nbe given. Then for every f ∈C_p^2N(N+N+3)(R^d)

|E(f(Sn(Z))−E(f(W)ΦN(W))|

≤ HNC_2(N+3)^2N(N+2N)(Z)(1 +C_2l

b

N(f)(Z))^2N+32^{(N+2)(lNb(f)+1)}L_Nb(f)× 1 n^N+12

(2.16)

in whichN = [N/2], Nb =N(2N+N+ 5), HN is a positive constant depending onN andW denotes a standard normal random variable in R^d.

As a consequence, takingf(x) =x^β with|β|=k, one gets

|E(S_n(Z)^β)−E(W^βΦ_N(W))| ≤ HNC_2(N+3)^2N(N+2N)(Z)(1 +C_2k(Z))^2N+32^(N+2)(k+1)× 1

n^N+12 . (2.17) 2.2 Basic decomposition and proof of the main result

LetN ∈ {0,1, . . .}.We define

T_N,r⁰ f(x) =

N+2X

l=1

1

l!D_r^(l)f(x). (2.18)

Since D_r^(l) ≡0 for l= 0,1,2, the above sum actually begins withl= 3 and of course this is the basic fact. Then, with the convention P2

l=3 = 0,we have T_N,r⁰ f(x) =

N+2X

l=3

1

l!D_r^(l)f(x).

We also define

T_N,r^1,Zf(x) = 1 (N + 2)!

X

|α|=N+3

Z 1

0

(1−λ)^N+2E(∂αf(x+λZr)Z_r^α)dλ and T_N,r¹ f(x) = T_N,r^1,Zf(x)−T_N,r^1,Gf(x).

(2.19)

For a matrixσ∈ Md×dwe recall the Laplace operatorL_σ associated toσ in (see (2.6)) and we define h⁰_N,σf(x) =f(x) +

XN

l=1

(−1)^l

2^ll! L^l_σf(x), with N = [N/2], (2.20) h¹_N,σf(x) = (−1)^N+1

2^N+1N! Z 1

0

s^NE(L^N+1_σ f(x+σ^1/2√

s W))ds. (2.21)

In (2.21),W stands for a standard Gaussian random variable. Then we define

U_N,r⁰ f(x) =E(h⁰_N,σrf(x+G_r)) and U_N,r¹ f(x) =h¹_N,σrf(x). (2.22) We now put our problem in a semigroup framework. For a sequenceX_k,k≥1, of independent r.v.’s, for 1≤k≤pwe define

P_k,k^Xf(x) =f and for p > k≥1 then P_k,p^Xf(x) =E f

x+

p−1

X

i=k

X_i

. (2.23)

We useP_k,p^Z andP_k,p^G. By using independence, we have the semigroup and the commutative property:

P_k,p^X =P_r,p^XP_k,r^X =P_k,r^XP_r,p^X k≤r≤p. (2.24) Moreover, form= 1, . . . , N we denote

Q^(m)_N,r1,...,rm = X

Pm

i=1qi+Pm i=1q^′_i>0 qi, q_i^′∈ {0,1}

Ym

i=1

U_N,r^q′i _i Ym

j=1

T_N,r^qj _j and

R^(m)_N,k,n = X

k≤r1<···<rm≤n

P_r^G_m+1,nP_r^G_m−1+1,rm· · ·P_r^G_1+1,r2P_k,r^G₁Q^(m)_N,r

1,...,rm.

(2.25)

Notice that in the first sum above the conditionsq_i, q^′_i∈ {0,1} andq₁+· · ·+q_m+q₁^′+· · ·+q_m^′ >0 say that at least one of q_i, q^′_i, i= 1, ..., m is equal to one. We notice that the operators T_N,r¹ _i and U_N,σ¹

ri

represent “remainders” and they are supposed to give small quantities of ordern^−12(N+1). So the fact that at least oneq_i orq^′_i is non null means that the product (Qm

i=1U_N,r^q′i _i)(Qm

i=1T_N,r^qi _i) has at least one term which is a remainder (so is small), and consequentlyR^(m)_N,k,n is a remainder also.

Finally we define Q^(N_N,r⁺¹⁾_1,...,rN+1 =

NY+1

i=1

(T_N,r⁰ _i+T_N,r¹ _i) and R^(N+1)_N,k,n = X

k≤r1<···<rN+1≤n

P_r^Z_N+1+1,nP_r^G_N+1,rN+1· · ·P_r^G_1+1,r2P_k,r^G₁Q^(N_N,r⁺¹⁾_1,...,r

N+1

(2.26)

We are now able to give our first result:

Proposition 2.3. Let N ≥1 and let T_N,r⁰ , h⁰_N,σr, R_N,k,n^(m) , m= 1, . . . , N+ 1, be given through (2.18), (2.20), (2.25), (2.26). Then for every 1≤k≤n+ 1and f ∈Cp^N(2N+N+3)(R^d) one has

P_k,n+1^Z f =P_k,n+1^G f+ Xn

m=1

X

k≤r1<···<rm≤n

P_k,n+1^G Y^m

i=1

T_N,r⁰ _iY^m

j=1

h⁰_N,σ

rj

f+

NX+1

m=1

R^(m)_N,k,nf. (2.27) Proof. Step 1 (Lindeberg method) We use the Lindeberg method in terms of semigroups: for 1≤k≤n+ 1

P_k,n+1^Z −P_k,n+1^G = Xn

r=k

P_r+1,n+1^Z (P_r,r+1^Z −P_r,r+1^G )P_k,r^G. Then we define

A_k,p=1_{1≤k≤p−1≤n}(P_p^Z_−1,p−P_p^G_−1,p)P_k,p^G₋₁ (2.28) and the above relation reads

P_k,n+1^Z =P_k,n+1^G + Xn

r=k

P_r+1,n+1^Z A_k,r+1. (2.29)

We will write (2.29) as a discrete time Volterra type equation (this is inspired from the approach to the parametrix method given in [9]: see equation (3.1) there). For a family of operators F_k,p, k≤p we defineAF by

(AF)_k,p =

p−1

X

r=k

F_r+1,pA_k,r+1 and we write (2.29) in functional form:

P^Z =P^G+AP^Z. (2.30)

By iteration,

P^Z=P^G+AP^G+· · ·+A^NP^G+A^N+1P^Z. (2.31) By the commutative property in (2.24), straightforward computations give

(A^mP^G)_k,p=1_k≤p−m^X

k≤r1<···<rm≤p−2

P_r^G_m+1,p−1P_r^G_m−1+1,rm· · ·P_r^G_1+1,r2P_k,r^G₁(P_p^Z_−1,p−P_p^G_−1,p)×

×(P_r^Z_m,rm+1−P_r^G_m,rm+1)(P_r^Z_{m−1,rm−1+1}−P_r^G_{m−1,rm−1+1})· · ·(P_r^Z_1,r1+1−P_r^G_1,r1+1).

(2.32) Step 2 (Taylor formula) The drawback of (2.31) is thatA depends onP^Z also, see (2.28). So, we use now the Taylor’s formula in order to eliminate this dependance. We use (2.4) and we consider a

Taylor approximation at the level of an error of ordern^−N+22 . We use the following expression for the Taylor’s formula: for f ∈C_p^∞(R^d),

f(x+y) =f(x) +

N+2X

p=1

1 p!

X

|α|=p

∂_αf(x)y^α+ 1 (N + 2)!

X

|α|=N+3

y^α Z 1

0

(1−λ)^N+2∂_αf(x+λy)dλ

Then we have, withD^(l)_r defined in (2.7),

(P_r,r+1^Z −P_r,r+1^G )f(x) =E(f(x+Z_r))−E(f(x+G_r))

=

N+2X

l=1

1

l!D_r^(l)f(x) + 1 (N + 2)!

X

|α|=N+3

Z 1

0

(1−λ)^N+2

E(∂αf(x+λZr)Z_r^α)−E(∂αf(x+λGr)G^α_r) dλ

=T_N,r⁰ f(x) +T_N,r¹ f(x).

By using the independence property, one can apply commutativity and by using (2.32) we have (A^mF)_k,r+1 =1_k≤r+1−m ^X

k≤r1<···<rm≤r

F_rm+1,r+1P_r^G_m−1+1,rm· · ·P_r^G_1+1,r2P_k,r^G₁ Ym

j=1

(T_N,r⁰ _j+T_N,r¹ _j). (2.33)

Notice that the operator in (2.33) acts on f ∈C^m(N+3). In particular, (A^mP^G)_k,n+1=1_k≤n+1−m ^X

k≤r1<···<rm≤n

P_r^G_m+1,n+1P_r^G_m−1+1,rm· · ·P_r^G_1+1,r2P_k,r^G₁ Ym

j=1

(T_N,r⁰ _j +T_N,r¹ _j) (2.34) Step 3 (Backward Taylor formula)Since

P_r^G_m+1,n+1P_r^G_m−1+1,rm· · ·P_r^G_1+1,r2P_k,r^G₁f(x) =E f

x+ Xn

i=k

G_k− Xm

j=1

G_rj , the chain P_r^G_m+1,n· · ·P_r^G_1+1,r2P_k,r^G

1 contains all the steps, except for the steps corresponding to r_i, i= 1, ..., m(remark that for each i,P_r^G_i,ri+1 is replaced withT_N,r⁰ _i+T_N,r¹ _i). In order to “insert” such steps we use the backward Taylor formula (B.3) up to orderN = [N/2] (see next Appendix B). So, we take h⁰_N,σr and h¹_N,σr as in (2.20) and (2.21) respectively and we have

P_r^G_1+1,r2P_k,r^G₁f(x) = E f

x+

rX2−1

i=k

Gi−Gr1

= E h⁰_N,σr

1f x+

rX2−1

i=k

G_i +E

h¹_N,σr

1f x+

rX2−1

i=k

G_i−G_r1

= P_r^G_1+1,r2P_k,r^G₁(P_r^G_1,r1+1h⁰_N,σr

1 +h¹_N,σr

1)f(x)

= P_r^G_1+1,r2P_k,r^G₁(U_N,r⁰ ₁ +U_N,r¹ ₁)f(x),

U_N,r⁰ ₁ and U_N,r¹ ₁ being given in (2.22). We use this formula in (2.34) for every i= 1,2, ..., m and we get

(A^mP^G)_k,n+1= X

k≤r1<···<rm≤n

P_r^G_m+1,n+1· · ·P_r^G_1+1,r2P_k,r^G₁Y^m

i=1

(U_N,r⁰ _i+U_N,r¹ _i)Y^m

j=1

(T_N,r⁰ _j+T_N,r¹ _j)

. (2.35)

Notice that the above operator acts onCp^m(2N+N+5)(R^d). Our aim now is to isolate the principal term, that is the sum of the terms where onlyU_N,r⁰ _i and T_N,r⁰ _i appear. So we write

(A^mP^G)_k,n+1 = X

k≤r1<···<rm≤n

P_r^G_m+1,n+1· · ·P_r^G_1+1,r2P_k,r^G₁Y^m

i=1

U_N,r⁰ _iY^m

j=1

T_N,r⁰ _j

+ X

k≤r1<···<rm≤n

P_r^G_m+1,n+1· · ·P_r^G_1+1,r2P_k,r^G₁Q^(m)_N,r1,...,rm withQ^(m)_N,r

1,...,rm defined in (2.25). The second term is just R^(m)_N,k,n in (2.25). In order to compute the first one we notice that for every r^′ < r < r^′′ we have

P_r+1,r^G ′′P_r^G′,rP_r,r+1^G =P_r^G′,r^′′

so that

P_r^G_m+1,n+1· · ·P_r^G_1+1,r2P_k,r^G₁( Ym

i=1

U_N,r⁰ _i) =P_k,n+1^G ( Ym

i=1

h⁰_N,σ

ri).

Then, form= 1, ..., N

(A^mP^G)_k,n+1= X

k≤r1<···<rm≤n

P_k,n+1^G Y^m

i=1

h⁰_N,σ

ri

Y^m

i=1

T_N,r⁰ _i

+R^(m)_N,k,n. We treat nowA^N+1P^Z. Using (2.33) we get

(A^N+1P^Z)_k,n+1= X

k≤r1<...<rN+1≤n

P_r^Z_N+1+1,n+1P_r^G_N+1,rN+1...P_r^G_1+1,r2P_k,r^G₁ YN

i=1

(T_N,r⁰ _i +T_N,r¹ _i) =R^(N+1)_N,k,n, which acts onC_p^N(N+3).

We give now some useful representations of the remainders.

Lemma 2.4. Let m∈ {1, ..., N + 1} and r₁ <· · ·< r_m ≤n be fixed. Set N_m:=m(2N +N + 5) for m ≤ N and Nm = (N + 1)(N + 3) otherwise. Then, the operators Q^(m)_N,r1,...,rm defined in (2.25) for m= 1, . . . , N and in (2.26) for m=N+ 1, can be written as

Q^(m)_N,r

1,...,rmf(x) = X

3≤|α|≤Nm

a^r_n^1,...,rm(α)θ_r^α_1,...,rm∂_αf(x) (2.36)

where a_n(α)∈R are suitable coefficients with the property

|a^r_n^1,...,rm(α)| ≤ (CC₂^N+1(Z))^m

n^N+3m2 , (2.37)

andθ_r^α_1,...,rm:C_p^∞(R^d)→C_p^∞(R^d) is an operator which verifies

|θ_r^α_1,...,rm∂_αf(x)| ≤ 2^lNm(f)+1C_2(N^1/2₊₃₎(Z)(1 +C_2lNm(f)(Z))²m

L_Nm(f)(1 +|x|)^lNm(f) (2.38) C >0 being a suitable constant. Moreover, θ^α_r1,...,rm can be represented as

θ^α_r1,...,rmf(x) = Z

(Rd)^2m

f(x+y1+· · ·+y2m)µ^α_r1,...,rm(dy1, . . . , dy2m) (2.39) where µ^α_r1,...,rm is a finite signed measure.

Proof. In a first step we construct the measures µ^α_r1,...,rm and the operatorsθ_r^α_1,...,rm and in a second step we prove that the corresponding coefficients a^r_n^1,...,rm(α) verify (2.37). We start by representing T_N,r⁰ defined in (2.18). Set

ν_r^0,α(dy) =n^2l∆_α(r)δ₀(dy), |α|=l≥3.

Notice that if |α|=l≥3 then n^l2|∆α(r)| ≤2C_l(Z). So, we have T_N,r⁰ f(x) =

N+2X

l=3

1 n^2l

X

|α|=l

1 l!

Z

Rd

∂_αf(x+y)ν_r^0,α(dy) with Z

Rd

(1 +|y|)^γ|ν_r^0,α|(dy)≤2C_l(Z), |α|=l≤N+ 2.

(2.40)

Hereafter γ denotes a non negative power. ConcerningT_N,r¹ in (2.19), for |α|=N+ 3 set ν^1,α(dy) = n^N+32 R1

0(1−λ)^N+2(^y_λ)^α

µ_λZr(dy)−µ_λGr(dy)

dλ, that is ν^1,α(A) =n^N+32

Z 1

0

(1−λ)^N+2

E(Z_r^α1_λZr∈A)−E(G^α_r1_λGr∈A)

dλ, |α|=N + 3, for every Borel setA. Then we have

T_N,r¹ f(x) = 1 n^12(N+3)

X

|α|=N+3

1 (N+ 2)!

Z

Rd

∂_αf(x+y)ν_r^1,α(dy) with Z

Rd

(1 +|y|)^γ|ν_r^1,α|(dy)≤ 2^γ+1

N + 3C_2(N+3)^1/2 (Z)(1 +C_2γ(Z))^1/2, |α|=N + 3.

(2.41)

We represent now the operator U_N,r⁰ f(x) =E(h⁰_N,σrf(x+G_r)) with h⁰_N,σrf defined in (2.20). Notice that

h⁰_N,σr = XN

l=0

X

|α|=2l

c^σ_n^r(α)∂_α with c^σ_n^r(α) = (−1)^l 2^ll!

Yl

k=1

σr^{α2k−1,α2k}, |α|=l.

So, by denotingρ⁰_σr the law of G_r, we have U_N,r⁰ f(x) =E(h⁰_N,σrf(x+Gr)) =

XN

l=0

X

|α|=2l

c^σ_n^r(α) Z

Rd

∂αf(x+y)ρ⁰_σr(dy) with

|c^σ_n^r(α)| ≤ C2(Z)^l 2^ll!n^2l and

Z

Rd

(1 +|y|)^γ|ρ⁰_σr|(dy)≤2^γ(1 +C_γ(Z)).

(2.42)

We now obtain a similar representation forh¹_N,σf(x) defined in (2.21). Set ρ¹_σ(dy) = Z ¹

0

s^Nφ_σ1/2√

s W(y)ds dy, in which φ_σ1/2√

s W denotes the density of a centred Gaussian r.v. with covariance matrix sσ. Then we write

h¹_N,σf(x) = X

|α|=2(N+1)

b^σ_n(α) Z

Rd∂αf(x+y)ρ¹_σ(dy) with b^σ_n(α) = (−1)^N+1 2^N+1N!

NY+1

k=1

σ^{α2k−1,α2k}.