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 99f426aeb1" title="Click to view the MathML source">0<α≤2$0<\alpha \le 2$. Let ad72c258fc5e9a3e25627eb6032" 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 adda584dd0" title="Click to view the MathML source">≤$\le$, where 949b743490d849b1429b1556f5c8" title="Click to view the MathML source">d≥1$d\ge 1$ is a fixed integer. For e51141d38df41ad5">$n=(n_1,\dots ,n_d)\in {\mathbb{N}}^d$, define e9428c5e32a33c7f9ce56ca8e42">$\left|n\right|={\prod }_{i=1}^d{n}_i$. Let $\{X,X_n;n\in {\mathbb{N}}^d\}$ be a field of independent and identically distributed real-valued random variables. Set 94" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167715216301791-si9.gif">$S_n={\sum }_{k\le n}{X}_k$, e708c87584f09a317df94154e80658">$n\in {\mathbb{N}}^d$ and write 94b69a73f3ac736cb0dcf82d647070b3">$logx={log}_e(e\vee x),x\ge 0$. 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