A simple proof of heavy tail estimates for affine type Lipschitz recursions

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• 作者：Dariusz Buraczewski ; ^{dbura@math.uni.wroc.pl} ; Ewa Damek ^{edamek@math.uni.wroc.pl}
• 关键词：primary ; 60J05 ; 37Hxx
• 刊名：Stochastic Processes and their Applications
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：127
• 期：2
• 页码：657-668
• 全文大小：255 K
• 卷排序：127

文摘

We study the affine recursion Xn=AnXn−1+BnXn=AnXn−1+Bn where (An,Bn)∈R+×R(An,Bn)∈R+×R is an i.i.d. sequence and recursions Xn=Φn(Xn−1)Xn=Φn(Xn−1) defined by Lipschitz transformations such that Φ(x)≥Ax+BΦ(x)≥Ax+B. It is known that under appropriate hypotheses the stationary solution XX has regularly varying tail, i.e. limt→∞tαP[X>t]=C. However positivity of CC in general is either unknown or requires some additional involved arguments. In this paper we give a simple proof that C>0C>0. This applies, in particular, to the case when Kesten–Goldie assumptions are satisfied.