Closure property and maximum of randomly weighted sums with heavy-tailed increments

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• 作者：Yang Yang^a ; ^b ; ^c ; Remigijus Leipus^c ; ^d ; Jonas &Scaron ; iaulys^c ; ^{jonas.siaulys@mif.vu.lt" class="auth_mail}
• 关键词：62E20 ; 62H20
• 刊名：Statistics and Probability Letters
• 出版年：August 2014
• 年：2014
• 卷：91
• 期：Complete
• 页码：162-170
• 全文大小：385 K

文摘

In this paper, we consider the randomly weighted sum $S_2^{螛}={螛}_1{X}_1+{螛}_2{X}_2$, where the two primary random summands X₁$X_1$ and 13c7f554e6bcb89d073c421b51099c6" title="Click to view the MathML source">X₂$X_2$ are real-valued and dependent with long or dominatedly varying tails, and the random weights 螛₁${螛}_1$ and 螛₂${螛}_2$ are positive, with values in [a,b]$[a,b]$, 0<a≤b<∞$0<a\le b<\infty$, and arbitrarily dependent, but independent of X₁$X_1$ and X₂$X_2$. Under some dependence structure between X₁$X_1$ and X₂$X_2$, we show that 3c9160d01799fcc5fc995196c">$S_2^{螛}$ has a long or dominatedly varying tail as well, and obtain the corresponding (weak) equivalence results between the tails of $S_2^{螛}$ and $M_2^{螛}=max\{{螛}_1{X}_1,{螛}_1{X}_1+{螛}_2{X}_2\}$. As corollaries, we establish the asymptotic (weak) equivalence formulas for the tail probabilities of randomly weighted sums of even number of long-tailed or dominatedly varying-tailed random variables.