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_nR_n−1+q, n≥1, where R₀ is arbitrary, (M_n) 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 (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 P(M>1)>0.