Persistence of One-Dimensional AR(1)-Sequences

Günter Hinrichs (), Martin Kolb () and Vitali Wachtel ()
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Günter Hinrichs: Universität Augsburg
Martin Kolb: Universität Paderborn
Vitali Wachtel: Universität Augsburg

Journal of Theoretical Probability, 2020, vol. 33, issue 1, 65-102

Abstract: Abstract For a class of one-dimensional autoregressive sequences $$(X_n)$$(Xn), we consider the tail behaviour of the stopping time $$T_0=\min \lbrace n\ge 1: X_n\le 0 \rbrace $$T0=min{n≥1:Xn≤0}. We discuss existing general analytical approaches to this and related problems and propose a new one, which is based on a renewal-type decomposition for the moment generating function of $$T_0$$T0 and on the analytical Fredholm alternative. Using this method, we show that $$\mathbb {P}_x(T_0=n)\sim V(x)R_0^n$$Px(T0=n)∼V(x)R0n for some $$0

Keywords: Persistence; Quasistationarity; Autoregressive sequence; Primary 60J05; Secondary 60G40 (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-018-0850-0

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