Merging to semistable laws

Abstract

Показывается, что хотя асимптотические распределения в обыч ном смысле не существуют, функции распределения подходящим образом центрированных и нормированных частичных сумм неза висимых одинаково распределенных случайных величин из обла сти геометрического частичного притяжения равномерно «состремятся» с семейством полу устойчивых функций распределения. Бо лее того, даже и соответствующие, возможно, мягко урезанные, суммы состремятся. Аналогичный результат с состремлением также имеет силу для экстремальных наблюдений. Основной ре зультат иллюстрируется на примере обобщенных Петербургских игр. Ключевые слова и фразы: полуустойчивые законы, области гео метрического частичного притяжения, состремление, мягко урезан ные суммы, экстремальные наблюдения, обобщенные Петербург ские игры. 1. Semis table laws and their domains of geometric partial at traction. We begin with a probabilistic representation of infinitely divis ible real random variables and their «trimmed» versions from [7] to be used throughout along with the integral conventions: for a left-continuous function l(-) and a right-continuous function r(-) we write f^r(x) dl(x) = f[a,b) r (x) dl(x) and f a l(x) dr(x) = J (a b] l(x) dr(x) whenever these integrals make sense for the finite constants a < b as Lebesgue-Stieltjes integrals, in which case the usual formula holds for integration by parts. Let Ф be the class of all nonpositive, nondecreasing, right-continuous functions ip(-) defined on the positive half-line (0, oc) such that the integrals / е°° ^2(s) ds < oc for all e > 0. Let E[ j \ E { 2 j \ ..., j = 1,2, be two indepen dent sequences of independent exponentially distributed random variables with mean 1, and with the respective partial sums = E[ j^ + •-h

Merging to Semistable Processes

“…We shall use it to handle p-semistable distributions, 0 < p < 2, and the domain of the geometric partial attraction of a p-semistable distribution, see Megyesi [18], Csörgő and Megyesi [7].…”

“…The domain of geometric partial attraction of a semistable law was described by Grinevich and Khokhlov [13]. In Csörgő and Megyesi [7] the theory of semistable laws was studied in the framework of the 'probabilistic' approach of Csörgő [5] and Csörgő, Haeusler, and Mason [6]. In the case of distributions being in the domain of geometric partial attraction of a semistable law ordinary convergence in distribution takes place only along some subsequences.…”

“…It is the functional version of the merge theorem by Csörgő and Megyesi [7]. The accompanying laws in our theorem are distributions of semistable Lévy processes (random processes with stationary independent increments).…”

“…In the case of distributions being in the domain of geometric partial attraction of a semistable law ordinary convergence in distribution takes place only along some subsequences. However, a merge theorem is valid (Csörgő and Megyesi [7]). We say that two sequences {µ n } and {ν n } of probability measures are merging if lim n→∞ (µ n , ν n ) = 0 where is a distance of probability measures (the Lévy distance or the Prokhorov distance, say).…”

“…The accompanying laws in our theorem are distributions of semistable Lévy processes (random processes with stationary independent increments). As we want to prove the precise functional version of (a part of) Theorem 2 of [7], in the following sections we shall use the same centering and norming constants as the ones in [7].…”