Convergence rates and precise asymptotics for renewal counting processes and some first passage times

Abstract

Let X 1 , X 2 , . . . be i.i.d. random variables with mean 0 and common distribution function F , and set S n = X 1 +X 2 +...+X n , n ≥ 1. In recent years precise asymptotics as ε 0 have been proved for sums like ∞ n=1 n r/p−2 P (|S n | ≥ εn 1/p ). Our main results are analogs for renewal counting processes and first passage time processes of random walks. In the latter setup we consider the case E X > 0 as well as the case E X = 0.

Precise asymptotics in Spitzer and Baum–Katz's law of large numbers: the semistable case

“…The following result generalizes Theorem 1 in [17], Theorem 2 in [6] and Theorem 2.1 in [8] for the stable case to the semistable case.…”

Compression based homogeneity testing

“…Now, m(n) is the first exit time, when the AN sequence s i = (z j ) hits high level h n . Its AN is well known, see [3].…”

Precise asymptotics for a series of T. L. Lai

“…The literature on this so-called precise asymptotics problem is reasonably rich, almost exhaustive references being given in Spȃtaru [13]. (We record here three new papers in the field by Scheffler [11] and Rozovsky [9,10], and recent related work on counting processes, record times, and partial maxima: Gut and Steinebach [5], Gut [2], Wang and Yang [14], and Wang, Yan and Yang [15]. )…”