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 ≥ 1 , where R 0 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 .