i.i.d随机变量部分和最大值序列的强大数定律

Pub Date : 2019-06-10 DOI:10.19195/0208-4147.39.1.2

Shuhua Chang, Deli Li, A. Rosalsky

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引用次数: 0

摘要

设0 < p≤2，设{Xn;n≥1}为实值随机变量X的独立副本序列，设Sn = X1 +…+ Xn, n≥- 1。本文从Mikosch 1984的一个定理出发，建立了数列{max1≤k≤n |Sk|;N≥- 1}。更具体地说，给出了limn→∞max1≤k≤n |Sk|log n−1 = e1/p a.s，其中log x = loge max{e, x}， x≥- 0的充要条件。

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Strong laws of large numbers for the sequence of the maximum of partial sums of i.i.d. random variables

Let 0 < p ≤ 2, let {Xn; n ≥ 1} be a sequence of independent copies of a real-valued random variable X, and set Sn = X1 + . . . + Xn, n ≥ 1. Motivated by a theorem of Mikosch 1984, this note is devoted to establishing a strong law of large numbers for the sequence {max1≤k≤n |Sk| ; n ≥ 1}. More specifically, necessary and sufficient conditions are given forlimn→∞ max1≤k≤n |Sk|log n−1 = e1/p a.s.,where log x = loge max{e, x}, x ≥ 0.

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