Convergence to type I distribution of the extremes of sequences defined by random difference equation

Pawel Hitczenko

doi:10.1016/j.spa.2011.06.007

paper

Convergence to type I distribution of the extremes of sequences defined by random difference equation

arXiv:1106.4281 · doi:10.1016/j.spa.2011.06.007

Abstract

We study the extremes of a sequence of random variables $(R_{n})$ defined by the recurrence $R_{n} = M_{n} R_{n - 1} + q$ , $n \geq 1$ , where $R_{0}$ is arbitrary, $(M_{n})$ are iid copies of a non--degenerate random variable $M$ , $0 \leq M \leq 1$ , and $q > 0$ is a constant. We show that under mild and natural conditions on $M$ the suitably normalized extremes of $(R_{n})$ converge in distribution to a double exponential random variable. This partially complements a result of de Haan, Resnick, Rootzén, and de Vries who considered extremes of the sequence $(R_{n})$ under the assumption that $¶ (M > 1) > 0$ .

to appear in Stochastic Processes and their Applications