Exact Asymptotics in log log Laws for Random Fields

Convergence rates of the LIL for random fields in Hilbert spaces

“…It is easily seen that we extend the results of Gut and Spȃtaru [6] and Spȃtaru [10] to the Hilbert space setting.…”

“…After that, Gut and Spȃtaru [6] and Spȃtaru [10] considered the exact convergence rates for random fields, and Huang and Zhang [9] investigated the precise rates of the law of the logarithm in Hilbert space. Inspired by them, we aim to study the convergence rates of the law of the iterated logarithm for Hilbert-space-valued i.i.d.…”

Precise asymptotics for a series of T. L. Lai

“…The classical method, as illustrated in Heyde [6], Spȃtaru [12], Gut and Spȃtaru [3], etc., proceeds with the assumption that X is in the domain of attraction of a stable law, and uses a version of the powerful Fuk-Nagaev inequality (see Spȃtaru [12]). This inequality is applicable only if the working condition has the form E |X| r < ∞ for some r. The second approach, introduced and practised by Gut and Spȃtaru [4], and Spȃtaru [13], is suitable for cases when Fuk-Nagaev type inequalities prove useless. It assumes existence of finite variance, and after the first step, it goes on via truncation and departure from normality.…”

“…Because the Fuk-Nagaev inequality is inadequate, we make use of the second method. However, appealing to the Berry-Esseen inequality to estimate the error in the normal approximation, as in Gut and Spȃtaru [4], and Spȃtaru [13], does not suffice, so we need the non-uniform estimate of Nagaev (see, e.g., Petrov [8], p. 125). Nevertheless, since the factor n −1/2 in the Nagaev inequality (and in the Berry-Esseen inequality) is unimprovable, the case p ≥ 3/2 remains unsettled in general; cf.…”

“…Then it is sensible to look for a normalizing function g(ε), ε > α, such that the ratio f (ε)/g(ε) has a nondegenerate limit l as ε α. 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].…”