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 (Rn) defined by the recurrence Rn=MnRn-1+q, n>=1, where R0 is arbitrary, (Mn) are iid copies of a non-degenerate random variable M, 0<=M<=1, and q>0 is a constant. We show that under mild and natural conditions on M the suitably normalized extremes of (Rn) 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 (Rn) under the assumption that .