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 91&_mathId=si1.gif&_user=111111111&_pii=S0167715216301791&_rdoc=1&_issn=01677152&md5=3ab1ee11a8cf5e03aa5d5099f426aeb1" title="Click to view the MathML source">0<α≤2$0<\alpha \le 2$. Let 91&_mathId=si2.gif&_user=111111111&_pii=S0167715216301791&_rdoc=1&_issn=01677152&md5=47816ad72c258fc5e9a3e25627eb6032" title="Click to view the MathML source">N^d${\mathbb{N}}^d$ be the 91&_mathId=si3.gif&_user=111111111&_pii=S0167715216301791&_rdoc=1&_issn=01677152&md5=16828075b5e4793256e93ec8c12a4942" title="Click to view the MathML source">d$d$-dimensional lattice equipped with the coordinate-wise partial order 91&_mathId=si4.gif&_user=111111111&_pii=S0167715216301791&_rdoc=1&_issn=01677152&md5=85e446f3bb8a4a1a9b1dc3adda584dd0" title="Click to view the MathML source">≤$\le$, where 91&_mathId=si5.gif&_user=111111111&_pii=S0167715216301791&_rdoc=1&_issn=01677152&md5=ab13949b743490d849b1429b1556f5c8" title="Click to view the MathML source">d≥1$d\ge 1$ is a fixed integer. For 91&_mathId=si6.gif&_user=111111111&_pii=S0167715216301791&_rdoc=1&_issn=01677152&md5=eaa08a8d12281933e51141d38df41ad5">91-si6.gif">$n=(n_1,\dots ,n_d)\in {\mathbb{N}}^d$, define 91&_mathId=si7.gif&_user=111111111&_pii=S0167715216301791&_rdoc=1&_issn=01677152&md5=9cc51e9428c5e32a33c7f9ce56ca8e42">91-si7.gif">$\left|n\right|={\prod }_{i=1}^d{n}_i$. Let 91&_mathId=si8.gif&_user=111111111&_pii=S0167715216301791&_rdoc=1&_issn=01677152&md5=2983e15cbef84622bfde111c8538baa9">91-si8.gif">$\{X,X_n;n\in {\mathbb{N}}^d\}$ be a field of independent and identically distributed real-valued random variables. Set 91&_mathId=si9.gif&_user=111111111&_pii=S0167715216301791&_rdoc=1&_issn=01677152&md5=6a22cbd3f6c00bab1790c65a45d35bdc">91-si9.gif">$S_n={\sum }_{k\le n}{X}_k$, 91&_mathId=si10.gif&_user=111111111&_pii=S0167715216301791&_rdoc=1&_issn=01677152&md5=c8e708c87584f09a317df94154e80658">91-si10.gif">$n\in {\mathbb{N}}^d$ and write 91&_mathId=si11.gif&_user=111111111&_pii=S0167715216301791&_rdoc=1&_issn=01677152&md5=94b69a73f3ac736cb0dcf82d647070b3">91-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