An extension of a theorem of Mikosch

Yuye Zou and Xiangdong Liu

Statistics & Probability Letters, 2017, vol. 120, issue C, 81-86

Abstract: Let 0<α≤2. Let Nd be the d-dimensional lattice equipped with the coordinate-wise partial order ≤, where d≥1 is a fixed integer. For n=(n1,…,nd)∈Nd, define |n|=∏i=1dni. Let {X,Xn;n∈Nd} be a field of independent and identically distributed real-valued random variables. Set Sn=∑k≤nXk, n∈Nd and write logx=loge(e∨x),x≥0. This note is devoted to an extension of a strong limit theorem of Mikosch (1984). By applying an idea of Li and Chen (2014) and the classical Marcinkiewicz–Zygmund strong law of large numbers for random fields, we obtain necessary and sufficient conditions for lim supn|Sn|(log|n|)−1=Δlimm→∞sup|n|≥m|Sn|(log|n|)−1=e1/αalmost surely .

Keywords: Chover-type law of the iterated logarithm; Random fields; Strong law of large numbers; Independent and identically distributed random variables (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1016/j.spl.2016.09.015

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