Asymptotic Constants for Minimal Distance in the Cen- Tral Limit Theorem

In this paper, we generalize the asymptotic result of Esseen (1958) concerning the Wasserstein distance of order one in the mean central limit theorem to the Wasserstein distances of order r for r in ]1, 2].


Introduction and main results.
In this paper we continue the research started in Rio (2009), concerning the Wasserstein distances in the central limit theorem.Let Ω be a probability space, rich enough to generate the set of probability laws on IR × IR.Let d be a pseudodistance on the set of real-valued random variables, such that d(X , Y ) depends only on the law of (X , Y ).Then, according to Paul Lévy (see Note B in Fréchet (1950) for this fact) the minimal distance d associated to d is defined by d(µ, ν) = inf{d(X , Y ) : X ∼ µ, Y ∼ ν}, where the infimum is taken over all random vectors (X , Y ) with respective marginal laws µ and ν.When E = IR, r ≥ 1 and d(X , Y ) = X − Y r , we denote by W r the so defined minimal distance on the space r of probability laws with a finite absolute moment of order r.This distance is usually called Wasserstein distance of order r.The distances W r are homogeneous of degree 1.
Throughout the paper, X 1 , X 2 , . . . is a sequence of independent and identically distributed realvalued random variables with mean zero and finite positive variance.We set S n = X 1 +X 2 +• • •+X n and v n = Var S n .We denote by µ n the law of v −1/2 n S n and by γ v the normal law with mean 0 and variance v.In a recent paper, Rio (2009) proved that, for r in [1,2], An interesting problem is then to find the limit in (1.1).Let F n denote the distribution function of µ n and Φ denote the distribution function of the standard normal.
Esseen (1958) proved that where A p (µ 1 ) is some nonnegative explicit constant.In the specific case p = 1, Consequently Esseen's result gives the asymptotic constant in (1.1) for r = 1.Zolotarev (1964) provided the following representation.Let Y and U be independent random variables, with respective distributions the standard normal law and the uniform law over where σ is the standard deviation of X 1 , α 3 = σ −3 IE(X 3 1 ) and h is the span of the distribution of X 1 for lattice distributions and 0 otherwise.For r > 1, it is known since a long time (cf.Dall'Aglio (1956) and Fréchet (1957) for more about this) that and consequently W r (µ n , γ 1 ) = F n − Φ r in general.In the next two theorems we describe the asymptotic behaviour of W r (µ n , γ 1 ) as n tends to ∞, for r in [1,2].As for r = 1, this behaviour depends on whether or not X 1 has a lattice distribution.
Theorem 1.1.Let Y be a random variable with standard normal law.Set α 3 = (IE(X 2 1 )) −3/2 IE(X 3 1 ).Let r be any real in ]1, 2].If the distribution of X 1 is not a lattice distribution and if We now state the results for lattice distributions.We will consider either smoothed sums or unsmoothed sums.For lattice distributions taking values in the arithmetic progression {a + kh : k ∈ Z Z} (h being maximal), the smoothed sums Sn are defined by where U is a random variable with uniform distribution over [−1/2, 1/2], independent of S n .We denote by μn the law of v −1/2 n Sn .Theorem 1.2 gives the asymptotic constants for smoothed or unsmoothed sums.Theorem 1.2.Let X 1 , X 2 , . . .be centered and independent identically distributed lattice random variables with variance σ 2 , taking values in the arithmetic progression {a + kh : k ∈ Z Z} (h being maximal).Let Y and U be two independent random variables, with respective laws the standard normal law and the uniform law over Remark 1.1.The constants appearing here are represented as in Zolotarev (1964), and consequently (b) still holds for r = 1.Proceeding as in Esseen (1958) one can prove that (a) is true in the case r = 1.
In the specific case r = 2, the asymptotic constant can be computed more explicitely.Let us state the corresponding result.
Corollary 1.3.Let X 1 , X 2 , . . .be centered and independent identically distributed lattice random variables with variance σ 2 and finite moment of order 4, taking values in the arithmetic progression {a + kh : k ∈ Z Z} (h being maximal).Then, with the same notations as in Theorems 1.1 and 1.2, Remark 1.1.In the non lattice case, as shown by Theorem 1.1, the above result holds with h = 0.
2 Non lattice distributions or lattice distributions and smoothed sums.
In this section, we prove Theorem 1.1.for non lattice distributions and Theorem 1.2(a) for lattice distributions and smoothed sums.Dividing the random variables by the standard deviation of X 1 , we may assume that the random variables X 1 , X 2 , . . .satisfy IE(X 2 1 ) = 1.Let Fn denote the distribution function of μn .The first step is the result below of pointwise convergence, which is known as the Cornish-Fisher expansion.As pointed in Hall (1992, Theorem 2.4 and final comments to Chapter 2), the Cornish-Fisher expansion of first order does not need the Cramer condition.To be exhaustive, we give a proof of this result Lemma 2.1.If µ 1 is not a lattice distribution, then, for any u in ]0, 1[, which holds true for non lattice distributions, and which holds true for lattice distributions.Moreover these estimates hold uniformly in x as n tends to ∞.Let We start by proving that, for any positive A, uniformly in x over [−A, A], For n large enough, Ψ(x) lies in [Φ(−2A), Φ(2A)] for any x in [−A, A].Then, by the Taylor formula at order 2 applied at Φ(x) with the increment Hence sup which ensures that, uniformly in x over [−A, A], (2.4) uniformly in x over [−A, A].Consequently, by (2.3), there exists some sequence ( n ) n of positive reals converging to 0 such that which ensures that Finally, by (2.5) and (2.6), Lemma 2.1(a) holds true with δ = Φ(−A/2).
Proof of Theorem 1.1.Recall that, for laws µ and ν with respective distribution functions F and Since Φ −1 (u) has the standard normal distribution under the Lebesgue measure over [0, 1], Theorem 1.1 will follow from the above inequality if we prove that, for N large enough, Hence (2.7b) follows from (2.7a) via the triangle inequality.The proof of (2.7a) will be done via Theorem 6.1 in Rio (2009), which is based on estimates of Borisov, Panchenko and Skilyagina (1998) for smooth functions of S n .For F and G distribution functions on the real line, let for some positive constant C depending only on r.Now 1 0 which implies (2.7a).Hence Theorem 1.1 holds true.

Lattice distributions.
In this section we prove Theorem 1.2(b).Again we may assume that σ 2 = 1.Let From (2.7a), it is enough to prove that, for any δ in ]0, 1/2[, In order to prove (3.1) we will use Lemma 2.1(b).For any distribution function F and any real x, let F (x − 0) = lim t↑x F (t).Let x 0 be the smallest number in the lattice {n −1/2 (a + kh) : k ∈ Z Z} such that F n (x 0 − 0) ≥ δ.For any relative integer l, set x l = x 0 + lhn −1/2 and a l = F n (x l ).Then, for any l in Z Z such that 0 < a l−1 < a l < 1, we have . Furthermore, it can easily be proven that Throughout the sequel, let t l (u) = (1 − u)a l−1 + ua l .Let m be the largest integer such that F n (x m ) ≤ 1 − δ.It comes from Esseen's estimates that a l−1 < a l for any l in [1, m], for n large enough.Then, for l in [1, m] and u in [0, 1], by Lemma 2.1(b) together with (3.3), It follows that the integrals from δ to a 0 and from a m to 1 − δ tend to 0 as n tends to ∞. Hence Let
By (2.1) again this condition holds true for n large enough if u belongs to [Φ(−A/2), Φ(A/2)].The estimate (2.1) also implies that the jumps of F n are uniformly bounded by o(n −1/2 ).