Recurrence and transience of random difference equations in the critical case

IF 1.5 Q2 PHYSICS, MATHEMATICAL
G. Alsmeyer, A. Iksanov
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引用次数: 1

Abstract

For i.i.d. random vectors $(M_{1},Q_{1}),(M_{2},Q_{2}),\ldots$ such that $M>0$ a.s., $Q\geq 0$ a.s. and $\mathbb{P}(Q=0)<1$, the random difference equation $X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,\ldots$, is studied in the critical case when the random walk with increments $\log M_{1},\log M_{2}$ is oscillating. We provide conditions for the null-recurrence and transience of the Markov chain $(X_{n})_{n\ge 0}$ by inter alia drawing on techniques developed in the related article Alsmeyer et al (2017) for another case exhibiting the null-recurrence/transience dichotomy.
临界情况下随机差分方程的递归性和暂态性
对于i.i.d随机向量$(M_{1},Q_{1}),(M_{2},Q_{2}),\ldots$,如$M>0$ a.s., $Q\geq 0$ a.s.和$\mathbb{P}(Q=0)<1$,研究了随机差分方程$X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,\ldots$,在具有增量$\log M_{1},\log M_{2}$的随机游走振荡的临界情况下。我们为马尔可夫链$(X_{n})_{n\ge 0}$的零递归和瞬态提供了条件,除其他外,我们还利用了相关文章Alsmeyer等(2017)中开发的技术,用于另一种显示零递归/瞬态二分法的情况。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.30
自引率
0.00%
发文量
16
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