On the local limit theorems for psi-mixing Markov chains

Merlevède, Florence; Peligrad, Magda; Peligrad, Costel

doi:10.48550/arxiv.2006.13361

Cited by 1 publication

(1 citation statement)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We observe that logarithmic in V N growth of various non-perioidicity characteristics of individual summands are often used in the theory of local limit theorems (see e.g. [20,21,23]). We will see from the examples of Section 10 that this result is close to optimal.…”

Section: Introductionmentioning

confidence: 99%

Edgeworth expansions for independent bounded integer valued random variables

Dolgopyat¹,

Hafouta²

2020

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%