A local limit theorem for linear random fields

Abstract

In this article, we establish a local limit theorem for linear fields of random variables constructed from i.i.d. innovations each with finite second moment. When the coefficients are absolutely summable we do not restrict the region of summation. However, when the coefficients are only square-summable we add the variables on unions of rectangle and we impose regularity conditions on the coefficients depending on the number of rectangles considered. Our results are new also for the dimension 1, that is, for linear sequences of random variables. The examples include the fractionally integrated processes for which the results of a simulation study is also included.

Limit theorems for linear random fields with innovations in the domain of attraction of a stable law

“…In the case where lim sup n sup i∈Z d |b ni | = ∞, we impose that ρ n → 0. Now we verify condition (16) in Theorem 4.1. A part of the argument is similar to the proof of Theorem 2 in Shukri (1976).…”

A Berry–Esseen bound of order $\protect \frac{1}{\protect \sqrt{n}} $ for martingales

“…and it is well known that B 2 n has order n. Hence n has order 1/ n in this case. In the long memory case ∞ i =0 |a i | = ∞, if we assume (8), B 2 n has order n 3−2α 2 (n) (e.g., Wu and Min [14]) and sup i ≤n |b n,i | has order n 1−α (n) (see Beknazaryan et al [1] for upper bound and Fortune et al [5] for lower bound in the case d = 1). Hence in this case n also has order 1/ n. In either case the Berry-Esseen bound has order 1/ n if δ n = O(n −1/2 ).…”

A Berry-Esseen bound of order $ 1/\sqrt{n} $ for martingales

“…n has order n 3−2α l 2 (n) (e.g., Wu and Min [14]) and sup i≤n |b n,i | has order n 1−α l(n) (see Beknazaryan et al [1] for upper bound and Fortune et al [5] for lower bound in the case d = 1). Hence in this case ǫ n also has order 1/ √ n. In either case the Berry-Esseen bound has order 1/…”