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 lick to view the MathML source">0<α≤2$0<\alpha \le 2$. Let lick to view the MathML source">N^d${\mathbb{N}}^d$ be the lick to view the MathML source">d$d$-dimensional lattice equipped with the coordinate-wise partial order lick to view the MathML source">≤$\le$, where lick to view the MathML source">d≥1$d\ge 1$ is a fixed integer. For lineImage" height="18" width="150" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167715216301791-si6.gif">$n=(n_1,\&hel<font\; color="red">li</font>p;,n_d)\in {\mathbb{N}}^d$, define lineImage" height="22" width="87" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167715216301791-si7.gif">$\left|n\right|={\prod }_{i=1}^d{n}_i$. Let lineImage" height="21" width="105" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167715216301791-si8.gif">$\{X,X_n;n\in {\mathbb{N}}^d\}$ be a field of independent and identically distributed real-valued random variables. Set lineImage" height="19" width="94" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167715216301791-si9.gif">$S_n={\sum }_{k\le n}{X}_k$, lineImage" height="16" width="48" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167715216301791-si10.gif">$n\in {\mathbb{N}}^d$ and write lineImage" height="16" width="174" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167715216301791-si11.gif">$logx={log}_e(e\vee x),x\ge 0$. 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