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

We study the extremes of a sequence of random variables $(R_n)$ defined by the recurrence $R_n=M_n{R}_{n-1}+q$, $n\ge 1$, where $R_0$ is arbitrary, $(M_n)$ are iid copies of a non–degenerate random variable $M$, $0\le M\le 1$, 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, Rootz'en, and de Vries who considered extremes of the sequence $(R_n)$ under the assumption that $\mathrm{\P }(M>1)>0$.