Zero bias transformation and asymptotic expansions

www.imstat.org/aihp 2012, Vol. 48, No. 1, 258–281

DOI:10.1214/10-AIHP384

© Association des Publications de l’Institut Henri Poincaré, 2012

Zero bias transformation and asymptotic expansions

Ying Jiao

Laboratoire de probabilités et modèles aléatoires, Université Paris VII, Paris, France. E-mail:jiao@math.jussieu.fr Received 10 May 2010; revised 17 June 2010; accepted 15 July 2010

Abstract. LetWbe a sum of independent random variables. We apply the zero bias transformation to deduce recursive asymptotic expansions forE[h(W )]in terms of normal expectations, or of Poisson expectations for integer-valued random variables. We also discuss the estimates of remaining errors.

Résumé. SoitWune somme de variables aléatoires indépendants. On applique la transformation zéro biais pour obtenir de façon recursive des développements asymptotiques deE[h(W )]en terme d’espérances par rapport à la loi normale, ou à la loi de Poisson si les variables aléatoires sont à valeurs entières. On discute aussi les bornes des termes d’erreur.

MSC:60G50; 60F05

Keywords:Normal and Poisson approximations; Zero bias transformation; Stein’s method; Reverse Taylor formula; Concentration inequality

1. Introduction

The zero bias transformation has been introduced by Goldstein and Reinert [15] in the framework of Stein’s method.

By the fundamental works of Stein [24,25], we know that a random variable (r.v.) Z with mean zero follows the normal distributionN (0, σ²)if and only ifE[Zf (Z)] =σ²E[f(Z)]for any Borel functionf such that both sides of the equality are well defined. More generally, for any r.v.Xwith mean zero and finite varianceσ²>0, a r.v.X^∗is said to have the zero biased distribution ofXif the equality

E Xf (X)

=σ²E f

X^∗

(1) holds for any differentiable functionf such that (1) is well defined. ByStein’s equation

xf (x)−σ²f(x)=h(x)−Φ_σ(h), (2)

wherehis a given function andΦ_σ(h)denotes the expectation ofhunder the distributionN (0, σ²), E

h(X)

−Φ_σ(h)=E

Xf_h(X)−σ²f_h(X)

=σ²E f_h

X^∗

−f_h(X)

, (3)

wherefhis the solution of Stein’s equation and is given by f_h= 1

σ²φσ(x) _∞

x

h(t)−Φ_σ(h)

φ_σ(t)dt. (4)

An important remark is thatX^∗needs not be independent ofX([15], see also [13]). LetW=X₁+ · · · +X_nwhereX_i (i=1, . . . , n)are independent random variables with mean zero and varianceσ_i>0,σ_W² :=Var[W] =σ₁²+ · · · +σ_n². Goldstein and Reinert have proposed the constructionW^∗=W^{(I )}+X^∗_I whereW⁽ⁱ⁾:=W−Xi,X_i^∗is independent

ofW⁽ⁱ⁾, andI is a random index taking values in{1, . . . , n}and independent of(X₁, . . . , X_n, X^∗₁, . . . , X^∗_n)such that P(I=i)=σ_i²/σ_W². This construction of the zero bias transformation is similar to Lindeberg’s method except that we here consider an average of punctual substitutions ofX_i byX^∗_i, while in Lindeberg’s method,X_i’s are progressively substituted by a central normal r.v. with the same variance.

The asymptotic expansion of E[h(W )] is a classical topic related to the central limit theorems. Using Stein’s method, Barbour [2,3] has obtained a full expansion ofE[h(W )]for sufficiently regular functions. Compared to the classical Edgeworth expansion (see [21], Chapter V, also [22]), the results of [2] do not require that the distributions of Xi’s be smooth. However, as a price paid, we need some regularity conditions on the functionh. The result of [2] can also be compared to those in [17,18] using Fourier transform. The key point of Barbour’s method is a Taylor type formula with cumulant coefficients, which allows to write the differenceE[Wf (W )] −σ_W²E[f(W )]as a series containing the cumulants ofW. TheW-expectations are then replaced by the normal expectations in an iterative way until the desired order. It has been pointed out in [22] that the key formula of Barbour can also be obtained by Fourier transform.

The zero bias transformation has been used in [11] to obtain a first-order correction term for the normal approxi- mation ofE[h(W )], where the motivation was to find a rapid numerical method for large-sized credit derivatives. The function of interest is the so-calledcall functionin finance:h(x)=(x−k)₊wherekis a real number. Since suchh is only absolutely continuous, the functionfh is not regular enough to have a third-order derivative. To achieve the estimates, we used a conditional expectation technique, together with a concentration inequality due to Chen and Shao [8,9].

The difficulty for generalizing the result in [11] to obtain a full expansion ofE[h(W )]is thatW andW^∗−Ware not independent. If we consider the Taylor expansion off_h(W^∗)atW and then take the expectation, the terms of the formE[f_h^(l)(W )(W^∗−W )^k]appear, wheref^(l) denotes thelth derivative off. For the first-order expansion in [11], a conditional expectation argument allowed us to replaceE[f_h(W )(W^∗−W )]byE[f_h(W )]E[W^∗−W]and to put the covariance in the error term. However, in the higher-order expansions, the error term can no longer contain such covariances. An alternative way is to consider the Taylor expansion ofE[f_h(W^∗)−f_h(W )]atW⁽ⁱ⁾. SinceX^∗_i is independent ofW⁽ⁱ⁾, there is no crossing term. However, the expectations of the formE[f_h^(l)(W⁽ⁱ⁾)]appear, which makes it difficult to apply the recurrence procedure. To overcome this difficulty, we shall propose a so-called reverse Taylor formula which will enable us to replaceE[f_h^(l)(W⁽ⁱ⁾)]byE[f_h^(l)(W )]up to an error term.

The main result of this paper is an expansion formulaE[h(W )] =C_N(h)+e_N(h),C_N(h)being the principal term of theNth-order expansion ande_N(h) being the remaining error. We shall precise the conditions under which the expansion formula is well defined. In particular, in order to include some irregular functions, we shall consider the continuous and the jump parts ofhseparately. Moreover, we also need to show thate_N(h) is “small” enough as an error term. In our results,e_N(h)is given in a recursive way so that it is actually a linear combination of the remaining terms of Taylor and reverse Taylor expansions and can be thus estimated. A key ingredient for the estimates is a concentration inequality which provides an upper bound for P(a≤W≤b) involving some power(b−a)^α of the interval length(b−a)with 0< α≤1. This allows to obtain estimates under relatively mild moment conditions on Xi’s. It is interesting to point out that the same method can be used in a discrete setting to obtain a Poisson asymptotic expansion for integer-valued summand random variables.

The rest of the paper is organized as follows. In Section2, we give the normal expansion by using the reverse Taylor formula and we discuss the conditions onhand onX_i’s. Section3is devoted to error estimates. We present the Poisson case in Section4. Finally, some technical proofs are left inAppendix.

2. Expansions in normal case

2.1. Reverse Taylor formula

Let N be a positive integer,X andY be two independent random variables such that Y has up toNth moments, andf be anN times differentiable function such thatf^(k)(X)andf^(k)(X+Y )are integrable for anyk=0, . . . , N.

We use the notationm^(k)_Y :=E[Y^k]/k!. Denote byδ_N(f, X, Y )the error term in theNth-order Taylor expansion of E[f (X+Y )]centered atX, i.e.

δ_N(f, X, Y ):=E

f (X+Y )

− N k=0

m^(k)_Y E

f^(k)(X)

. (5)

Recall that for anyN≥1, δ_N(f, X, Y )= 1

(N−1)! ₁

0

(1−t )^N−1E

f^{(N )}(X+t Y )−f^{(N )}(X) Y^N

dt (6)

provided that the right-hand side is well defined.

The so-called reverse Taylor formula gives an expansion ofE[f (X)]using the expectations of functions ofX+Y and the moments ofY. We specify some notation and conventions. First of all, N∗:=N\ {0} denotes the set of strictly positive integers. For any integerd≥1 and anyJ=(jl)^d_l=1∈N^d_∗,|J|is defined asj1+ · · · +jd, andm^(J)_Y :=

m^(j_Y¹⁾· · ·m^(j_Y^d). By convention,N⁰_∗denotes the set{∅}of the empty vector,|∅| =0 andm⁽_Y^∅)=1. We note that in the right–hand side of the expansion formula (7), the variablesX+Y andY are not independent.

Proposition 2.1 (Reverse Taylor formula). With the above notation,the equality

E f (X)

=

d≥0

(−1)^d

J∈N^d∗,|J|≤N

m^(J)_Y E

f^(|J|)(X+Y )

+ε_N(f, X, Y ) (7)

holds,whereε_N(f, X, Y )is defined as

ε_N(f, X, Y )= −

d≥0

(−1)^d

J∈N^d∗,|J|≤N

m^(J)_Y δ_N−|J|

f^(|J|), X, Y

. (8)

Proof. We replaceE[f^(|J|)(X+Y )]on the right-hand side of (7) by

N−|J| k=0

m^(k)_Y E

f^(|J|+k)(X)

+δ_N−|J|

f^(|J|), X, Y

and observe that the sum of terms containingδvanishes withε_N(f, X, Y ). Hence the right-hand side of (7) equals

d≥0

(−1)^d

J∈N^d∗,|J|≤N

m^(J)_Y

N−|J| k=0

m^(k)_Y E

f^(|J|+k)(X) .

If we split the terms wherek=0 and where 1≤k≤N− |J|, the above formula can be written as

d≥0

(−1)^d

J∈N^d_∗,|J|≤N

m^(J)_Y E

f^(|J|)(X) +

d≥0

(−1)^d

J∈N^d_∗,|J|≤N

m^(J)_Y

N−|J| k=1

m^(k)_Y E

f^(|J|+k)(X)

. (9)

We make the index changesJ=(J, k)andu=d+1 in the second part, and observe that it is just

u≥1

(−1)^u−1

J∈N^u_∗,|J|≤N

m^(J_Y⁾E

f^(|J|)(X) .

Thus, the terms in the first and the second parts of (9) cancel out except the one whered=0 in the first part. So (9)

equalsE[f (X)], which proves the proposition.

2.2. Admissible function space

In this subsection, we precise the functions for which we can make theNth-order expansion. We need conditions on both the regularity and the increasing speed at infinity of the function.

Recall ([20], Chapter VI) that any functiongonRwhich is locally of finite variation can be uniquely decomposed into the sum of a function of pure jump and a continuous function locally of finite variation and vanishing at the origin.

That is,g=g_c+g_d whereg_cis called thecontinuous partofgandg_dis thepurely discontinuous part.

Letα∈(0,1]andp≥0 be two real numbers. For any functionf onR, the following quantity has been defined by Barbour in [2]:

fα,p:=sup

x =y

|f (x)−f (y)|

|x−y|^α(1+ |x|^p+ |y|^p). (10) The finiteness of fα,p implies that the functionf is locally α-Lipschitz, and that the increasing speed off at infinity is at most of order|x|^α+p. All functionsf such thatfα,p<+∞form a vector space overR, and · α,p

is a norm on it. We list below several properties of · α,pand we leave the proofs in AppendixA.

Lemma 2.2. Letf be a function onR,α, β∈(0,1]andp, q≥0.

(1) Ifp≤q,thenfα,p<+∞impliesfα,q<+∞. (2) Ifα≤β,thenfβ,p<+∞impliesfα,p+β−α<+∞.

(3) IfP is a polynomial of degreed,thenfα,p<+∞impliesPfα,p+d<+∞.

(4) Assume thatF is a primitive function off,thenfα,p<+∞impliesF1,p+α<+∞. (HenceFα,p+1<

+∞by2.)

Inspired by [2], we introduce the following function space.

Definition 2.3. LetN≥0be an integer,andα∈(0,1],p≥0be two real numbers.Denote byH^Nα,pthe vector space of all Borel functionshonRverifying the following conditions:

(a) hhasNth derivative which is locally of finite variation and has finitely many jumps, (b) the continuous part ofh^{(N )}satisfiesh^{(N )}_c α,p<+∞.

Condition (a) implies that the pure jump part of h^{(N )} is bounded. Condition (b) implies that h^{(N )}_c has at most polynomial increasing speed at infinity, and so doesh. These conditions allow us to include some irregular functions such as the indicator functions. Letkbe a real number andI_k(x)=1_{x≤k}. ThenI_kα,pis clearly not finite. However, I_k,cα,p=0, which means that for anyα∈(0,1]and anyp≥0,I_k(x)∈H⁰α,p. Note that any functionhinHα,p⁰

can be decomposed as h=h_c+h_d, whereh_c satisfiesh_cα,p<+∞andh_d is a linear combination of indicator functions of the form1_{x≤k}plus a constant (so thathc(0)=0).

Proposition 2.4. LetN≥0be an integer,α, β∈(0,1]andp, q≥0be real numbers.Then the following assertions hold:

(1) whenN≥1,h∈H^N_α,pif and only ifh∈H^N_α,p⁻¹;

(2) ifp≤q,thenH^Nα,p⊂H^Nα,q;ifα≤β,thenH_β,p^N ⊂H^N_α,p+β−α; (3) whenN≥1,H^N_α,p⊂H^N_1,α⁻₊¹_p⊂H_α,p^N−₊¹₁;

(4) ifh∈H^N_α,pand ifP is a polynomial of degreed,thenP h∈H^N_α,p+d.

Proof. Assertion (1) results from the definition. Assertions (2)–(4) are consequences of Lemma2.2.

The following result is important and its proof which is rather technical is postponed to AppendixB.

Proposition 2.5. Ifh∈H^N_α,p,then the solution of Stein’s equation satisfiesf_h∈H^N_α,p⁺¹.

2.3. Normal expansion

We consider a family of independent random variablesXi(i=1, . . . , n)with mean zero and finite varianceσ_i²>0.

LetW=X₁+· · ·+X_nandσ_W² =Var(W ). Denote byX^∗_i a random variable which follows the zero-biased distribution ofX_iand is independent ofW⁽ⁱ⁾:=W−X_i.

Theorem 2.6. LetN≥0be an integer,α∈(0,1]andp≥0.Assume thath∈H^Nα,p.LetX₁, . . . , X_nbe independent random variables with mean zero and up to(N+max(α+p,2))th moments.ThenE[h(W )]can be written as the sum of two termsCN(h)andeN(h)whereC0(h)=Φσ_W(h)ande0(h)=E[h(W )] −Φσ_W(h),and recursively forN≥1,

C_N(h)=C₀(h)+ n

i=1

σ_i²

d≥1

(−1)^d−1

J∈N^d∗,|J|≤N

m^(J_X^◦)

i

m^(J_X^†_∗⁾

i −m^(J_X^†)

i

C_N−|J|

f_h^(|J|+1)

, (11)

e_N(h)= n i=1

σ_i²

d≥1

(−1)^d−1

J∈N^d∗,|J|≤N

m^(J_X^◦)

i

m^(J_X^†_∗⁾

i −m^(J_X^†)

i

e_N−|J|

f_h^(|J|+1) +

N k=0

ε_N−k

f_h^(k+1), W⁽ⁱ⁾, X_i m^(k)_X∗

i +δ_N

f_h, W⁽ⁱ⁾, X^∗_i

, (12)

where for any integerd≥1,and anyJ∈N^d_∗, J^†∈N∗denotes the last coordinate ofJ,andJ^◦denotes the vector in N^d_∗⁻¹obtained fromJby omitting the last coordinate.

Remark 2.7. The moments ofX^∗_i in(11)and (12)can be written as moments of X_i.By definition of the zero bias transformation,for anyk∈Nsuch thatX_i^∗haskth moment,one has

E X^∗_ik

= E[X^k_i⁺²] σ_i²(k+1). Hencem^(J_X^†_∗⁾

i =σ_i⁻²(J^†+2)m^(J_X^†+2)

i whereJ^†is the strictly positive integer figuring in the last coordinate ofJ.

Proof. We first prove that all terms in (11) and (12) are well defined. WhenN=0,h∈H⁰α,pandh(x)=O(|x|^α+p).

Hence E[h(W )] and Φσ_W(h) are well defined. Assume that we have proved for 0, . . . , N −1. Let h∈H^Nα,p. Then by Proposition2.4, h(x)∈H^Nα,p ⊂H_α,p^N−₊¹₁⊂ · · · ⊂H⁰_α,p+N, so h(x)=O(|x|^α+p+N). By Proposition 2.5, f_h∈H^Nα,p⁺¹and by Proposition2.4(1), for any|J| =1, . . . , N,f_h^(|J|+1)∈H^Nα,p^−|J|. So the induction hypothesis implies thatC_N−|J|(f_h^(|J|+1))ande_N−|J|(f_h^(|J|+1))are well defined. Furthermore, sincef_h^(k+1)(x)=O(|x|^α+p+N−k)for any k=0, . . . , N, the termsε_N−kandδ_N in (12) are well defined. Finally, all moments figuring in (11) and (12) exist.

We then prove the equalityE[h(W )] =C_N(h)+e_N(h). In the caseN =0, the equality holds by definition. In the following, we assume that the equalityE[h(W )] =C_k(h)+e_k(h)has been verified for anyk=0, . . . , N−1. By Stein’s equation (3),E[h(W )] −C₀(h)is equal to

σ_W²E f_h

W^∗

−f_h(W )

= n i=1

σ_i² E

f_h

W⁽ⁱ⁾+X^∗_i

−E

f_h(W )

. (13)

Consider the following Taylor expansion E

f_h

W⁽ⁱ⁾+X^∗_i

= N k=0

m^(k)_X∗

iE f_h^(k+1)

W⁽ⁱ⁾ +δN

f_h, W⁽ⁱ⁾, X_i^∗ .

By replacing E[f_h^(k+1)(W⁽ⁱ⁾)] in the above formula by its (N −k)th reverse Taylor expansion, we obtain that E[f_h(W⁽ⁱ⁾+X^∗_i)]equals

N k=0

m^(k)_X∗

i

d≥0

(−1)^d

J∈N^d∗|J|≤N−k

m^(J)_X

iE

f_h^(|J|+k+1)(W )

+ε_N−k

f_h^(k+1), W⁽ⁱ⁾, X_i +δ_N

f_h, W⁽ⁱ⁾, X_i^∗ .

Note that the term with indexesk=d=0 in the sum inside the bracket isE[f_h(W )]. ThereforeE[f_h(W⁽ⁱ⁾+X^∗_i)] − E[f_h(W )]can be written as the sum of the following three parts

n k=1

m^(k)_X∗

i

d≥0

(−1)^d

J∈N^d∗,|J|≤N−k

m^(J)_X

iE

f_h^(|J|+k+1)(W )

, (14)

d≥1

(−1)^d

J∈N^d_∗,|J|≤N

m^(J)_X

iE

f_h^(|J|+1)(W )

, (15)

N k=0

m^(k)_X∗

i

ε_N−k

f_h^(k+1), W⁽ⁱ⁾, X_i +δ_N

f_h, W⁽ⁱ⁾, X^∗_i

. (16)

By interchanging summations and then making the index changesK=(J, k)andu=d+1, we obtain (14)=

u≥1

(−1)^u−1

K∈N^u∗,|K|≤N

m^(K_X^◦)

i m^(K_X∗^†)

i E

f_h^(|K|+1)(W ) .

As the equalitym^(J)_X

i =m^(J_X^◦)

i m^(J_X^†)

i holds for anyJ, (14)+(15) equals

d≥1

(−1)^d−1

J∈N^d∗,|J|≤N

m^(J_X^◦)

i

m^(J_X^†_∗⁾

i −m^(J_X^†)

i

E

f_h^(|J|+1)(W ) .

By the hypothesis of induction, we have E[f_h^(|J|+1)(W )] =C_N−|J|(f_h^(|J|+1))+e_N−|J|(f_h^(|J|+1)). So the equality

E[h(W )] =C_N(h)+e_N(h)follows.

Denote byμ^(k)_i thekth moment ofX_i. We give the first few explicit expansion formulas below:

C1(h)=Φσ_W(h)+C0

f_h⁽²⁾ⁿ

i=1

μ⁽³⁾_i 2 ; C₂(h)=Φ_σW(h)+C₁

f_h⁽²⁾ⁿ

i=1

μ⁽³⁾_i 2 +C₀

f_h⁽³⁾ⁿ

i=1

μ⁽⁴⁾_i

6 −(μ⁽²⁾_i )² 2

; C₃(h)=Φ_σW(h)+C₂

f_h⁽²⁾ⁿ

i=1

μ⁽³⁾_i 2 +C₁

f_h⁽³⁾ⁿ

i=1

μ⁽⁴⁾_i

6 −(μ⁽²⁾_i )² 2

+C₀

f_h⁽⁴⁾ⁿ

i=1

μ⁽⁵⁾_i

24 −5μ⁽²⁾_i μ⁽³⁾_i 12

.

Note that inCN(h), there appear normal expectations of the operators which are of the formh→f_h^(l). As pointed out by Barbour [2], p. 294, such expectations can be expressed as those ofhmultiplied by a Hermite polynomial, see also remark below.

Remark 2.8. Let us compare our result with Barbour’s.In the main theorem of[2],the condition on the functionhfor theNth-order expansion is:hisN times differentiable andh^{(N )}α,p<∞.The conditionh∈Hα,p^N in Theorem2.6 is weaker since we separate the continuous and the jump parts ofh.An example inH¹α,p which does not satisfy the

condition of[2]ish(x)=(x−k)⁺sincehα,pis infinite.Concerning the summand variables,up to(N+max(α+ p,2))th moments ofX_i’s are needed for theNth expansion and up to(N+α+p+2)th moments,which are similar to[2],are required to achieve the error estimates(see Section3.2for details).We consider now the expansion formula.

Letr≥1be an integer.By Remark2.7and the fact thatX_i’s are with mean zero,the moment term figuring in(11) satisfies

σ_i²

d≥1

(−1)^d−1

J∈N^d_∗,|J|=r

m^(J_X^◦)

i

m^(J_X^†_∗⁾

i −m^(J_X^†)

i

=

d≥1

(−1)^d−1

J∈N^d_∗,|J|=r

m^(J_X^◦)

i

J^†+2 m^(J_X^†+2)

i −2m^(J_X^†)

i m⁽²⁾_X

i

=

d≥1

(−1)^d−1

J∈N^d_∗,|J|=r

J^†+2 m^(J_X^◦)

i m^(J_X^†+2)

i −

d≥1

(−1)^d−1

J∈N^d_∗,|J|=r

2m^(J_Xi^◦)m^(J_X^†)

i m⁽²⁾_X

i

=

d≥1

(−1)^d−1

K∈N^d∗,|K|=r+2 K^†≥3

K^†m^(K)_X

i +

d≥1

(−1)^d−1

K∈N^d∗,|K|=r+2 K^†=2

K^†m^(K)_X

i

=

K∈

d≥1N^d∗

|K|=r+2

(−1) ^(K)−1K^†m^(K)_X

i ,

where in the third equality,we use the index changesK=(J^◦,J^†+2)for the first term andK=(J,2),d=d+1for the second term,and in the last equality, (K)denotes the length of the vectorK.Denote byκ_i^(r)therth cumulant of X_i.By the recurrence formula which relates cumulants and moments(e.g., [23]):

κ_i^(r)=μ^(r)_i −

r−1

j=1

r−1 j

μ^{(j )}_i κ_i^{(r−j )}, (17)

we obtain

σ_i²

d≥1

(−1)^d−1

J∈N^d∗,|J|=r

m^(J_X^◦)

i

m^(J_X^†_∗⁾

i −m^(J_X^†)

i

= κ_i^(r+2)

(r+1)!. (18)

In fact,the formula(17)can also be written as κ_i^(r)

(r−1)!=rm^(r)_X

i −

r−1

j=1

m^{(j )}_X

i

κ_i^{(r−j )} (r−j−1)!. Denote byη_i^(r)the sum

|K|=r(−1) ^(K)−1K^†m^(K)_X

i .A direct verification showsη_i⁽¹⁾=κ_i⁽¹⁾=0and for anyr≥2, rm^(r)_X

i −

r−1

j=1

m^{(j )}_X

iη_i^{(r−j )}=rm^(r)_X

i −

r−1

j=1

m^{(j )}_X

i

|K|=r−j

(−1) ^(K)−1K^†m^(K)_X

i

=rm^(r)_X

i −

r−1

j=1

|K|=r−j

(−1) ^(K)−1K^†m^(j,K)_X

i

=rm^(r)_X

i +

|J|=r, (J)≥2

(−1) ^(J)−1J^†m^(J)_X

i =η^(r)_i ,

whereJ=(j,K)in the third equality.By induction,we obtainη^(r)_i =κ_i^(r)/(r−1)!for anyr∈N_∗and hence(18).So the expansion formula(11)becomes

C_N(h)=C₀(h)+ N r=1

C_N−r

f_h^(r+1)κ^(r+2)(W )

(r+1)! , (19)

whereκ^(r)(W )denotes therth cumulant ofW.We thus recover the result of Barbour. 3. Estimates of remaining errors

3.1. Concentration inequalities

We shall prove a concentration inequality which gives an upper bound for P(a≤W≤b),a andb being two real numbers. The special point here is to consider a parameterα≤1 and to give a bound depending sub-linearly onb−a.

The case whereα=1 has been studied in [8,9,11,14]. The following result, measuring the nearness betweenXand X^∗, is useful to prove the concentration inequality.

Lemma 3.1. Letα∈(0,1]andXbe a r.v.with mean zero,finite varianceσ²>0and up to(α+2)th moments.Let X^∗have the zero biased distribution ofXand be independent ofX.Then,for anyε >0,

PX−X^∗> ε

≤ 1

2ε^α(α+1)σ²EX^sα+2 ,

whereX^s=X−XandXis an independent copy ofX.

Proof. Similarly to the Markov inequality, the following inequality holds:

PX−X^∗> ε

≤ 1

ε^αEX−X^∗α .

For any locally integrable even functionf, by definition of the zero bias transformation and the fact thatXandX^∗ are independent,

E f

X^∗−X

= 1 2σ²E

X^sF X^s

, whereF (x)=_x

0 f (t )dt. In particular, EX−X^∗α

= 1

2(α+1)σ²EX^sα+2 ,

which implies the lemma.

Lemma 3.2. We keep the notation of Theorem2.6and suppose thatf:R→Randg:R²→Rare two functions such that the variance off (W )andg(X_i, X^∗_i)exist fori=1, . . . , n.Then

Cov

f (W ), g

XI, X_I^∗≤Var[f (W )]^1/2 σ_W²

_n

i=1

σ_i⁴Var g

Xi, X_i^∗1/2

. (20)

In particular,for anyε≥0, Cov[1_{a≤W≤b},1_{|XI−X∗

I|≤ε}]≤1 4

_n

i=1

σ_i⁴ σ_W⁴

1/2

. (21)