Edgeworth expansions for independent bounded integer valued random variables

Abstract

We obtain asymptotic expansions for local probabilities of partial sums for uniformly bounded independent but not necessarily identically distributed integer-valued random variables. The expansions involve products of polynomials and trigonometric polynomials. Our results do not require any additional assumptions. As an application of our expansions we find necessary and sufficient conditions for the classical Edgeworth expansion. It turns out that there are two possible obstructions for the validity of the Edgeworth expansion of order r. First, the distance between the distribution of the underlying partial sums modulo some h ∈ N and the uniform distribution could fail to be o(σ 1−r N ), where σ N is the standard deviation of the partial sum. Second, this distribution could have the required closeness but this closeness is unstable, in the sense that it could be destroyed by removing finitely many terms. In the first case, the expansion of order r fails. In the second case it may or may not hold depending on the behavior of the derivatives of the characteristic functions of the summands whose removal causes the break-up of the uniform distribution. We also show that a quantitative version of the classical Prokhorov condition (for the strong local central limit theorem) is sufficient for Edgeworth expansions, and moreover this condition is, in some sense, optimal.

Edgeworth expansions for integer valued additive functionals of uniformly elliptic Markov chains

“…) does not imply that the Edgeworth expansion holds even in the independent case, see [6,Example 10.2]. In fact, in the independent case if one of the Y n 's is uniformly distributed modulo m then E[e itS N ] = 0 for all N large enough and for every nonzero resonant point of the form t = 2πl m .…”

“…In fact, in the independent case if one of the Y n 's is uniformly distributed modulo m then E[e itS N ] = 0 for all N large enough and for every nonzero resonant point of the form t = 2πl m . However, this does not imply that the derivatives of the characteristic function of S N − E[S N ] vanish at t, and hence by [6,Theorem 1.5] expansions of an arbitrary order r might not hold.…”

“…Indeed, we assume that the sums of the variances of X 2n , n ≤ N/2 is lager than the sum of corresponding sum along the odd indexes, and then condition on the odd indexes. Now we can apply [6,Lemma 3.3].…”

“…During the 20th century, the work of many authors led to the development of asymptotic expansions in both the CLT and the LLT, see [12,10] and references therein for more details. Recently in [6], for bounded independent random variables Y n we gave a complete characterization for expansions of an arbitrary order r by means of the rate of decay of the characteristic function of S N − E[S N ] and their first r − 1 derivatives at nonzero "resonant points" of the form t = 2πl m with 0 < m ≤ 2K and 0 ≤ l < m, where K = sup n Y n L ∞ . The main probabilistic interpretation of these results was a characterization of "super stable" Edgeworth expansions of an arbitrary order r using only the decay rates of the distance of the distribution of S N − ℓ j=1 Y k j,N modulo m from the uniform distribution, for all m, for an arbitrary choice of indexes k 1,N , ..., k ℓ,N .…”

“…In this paper we will characterize a certain type of super stable Edgeworth expansions of an arbitrary order r (see the precise definition below). That is, we will generalize [6,Theorem 1.8] to integer valued additive functionals of uniformly elliptic Markov chains.…”