Recurrence and transience of random difference equations in the critical case

2021/05/11 by Alsmeyer, Gerold, Iksanov, Alexander
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2105.04994

Abstract

For i.i.d. random vectors (M₁,Q₁),(M₂,Q₂),… such that M>0 a.s., Q≥ 0 a.s. and ℙ(Q=0)<1, the random difference equation X_n=M_nX_n-1+Q_n, n=1,2,…, is studied in the critical case when the random walk with increments log M₁,log M₂ is oscillating. We provide conditions for the null-recurrence and transience of the Markov chain (X_n)_{n≥ 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.

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