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 8cf5e03aa5d5099f426aeb1" title="Click to view the MathML source">0<α≤2$0<\alpha \le 2$. Let 8fc5e9a3e25627eb6032" title="Click to view the MathML source">N^d${\mathbb{N}}^d$ be the 8c12a4942" title="Click to view the MathML source">d$d$-dimensional lattice equipped with the coordinate-wise partial order ≤$\le$, where d≥1$d\ge 1$ is a fixed integer. For e51141d38df41ad5">$n=(n_1,\dots ,n_d)\in {\mathbb{N}}^d$, define 9cc51e9428c5e32a33c7f9ce56ca8e42">$\left|n\right|={\prod }_{i=1}^d{n}_i$. Let $\{X,X_n;n\in {\mathbb{N}}^d\}$ be a field of independent and identically distributed real-valued random variables. Set $S_n={\sum }_{k\le n}{X}_k$, 8c87584f09a317df94154e80658">$n\in {\mathbb{N}}^d$ and write b0dcf82d647070b3">$logx={log}_e(e\vee x),x\ge 0$. 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