An extension of a theorem of Mikosch

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• 作者：Yuye Zou^a ; Xiangdong Liu^b ; ^{tliuxd@jnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60F15 ; 60G50
• 刊名：Statistics & Probability Letters
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：120
• 期：Complete
• 页码：81-86
• 全文大小：383 K

文摘

Let 0<α≤2$0<\alpha \le 2$. Let e9a3e25627eb6032" title="Click to view the MathML source">N^d${\mathbb{N}}^d$ be the e93ec8c12a4942" title="Click to view the MathML source">d$d$-dimensional lattice equipped with the coordinate-wise partial order bb8a4a1a9b1dc3adda584dd0" title="Click to view the MathML source">≤$\le$, where 849b1429b1556f5c8" title="Click to view the MathML source">d≥1$d\ge 1$ is a fixed integer. For $n=(n_1,\dots ,n_d)\in {\mathbb{N}}^d$, define 9cc51e9428c5e32a33c7f9ce56ca8e42">$\left|n\right|={\prod }_{i=1}^d{n}_i$. Let 84622bfde111c8538baa9">$\{X,X_n;n\in {\mathbb{N}}^d\}$ be a field of independent and identically distributed real-valued random variables. Set $S_n={\sum }_{k\le n}{X}_k$, e708c87584f09a317df94154e80658">$n\in {\mathbb{N}}^d$ and write $logx={log}_e(e\vee x),x\ge 0$. 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