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)(Rn) defined by the recurrence Rn=MnRn−1+qRn=MnRn−1+q, n≥1n≥1, where R0R0 is arbitrary, (Mn)(Mn) are iid copies of a non-degenerate random variable MM, 0≤M≤10≤M≤1, and q>0q>0 is a constant. We show that under mild and natural conditions on MM the suitably normalized extremes of (Rn)(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)(Rn) under the assumption that P(M>1)>0P(M>1)>0.

► We study extremes of sequences defined by random linear equations in the case where the limiting random variables have light tails.
► We show that under natural and mild conditions, properly normalized extremes converge to a double-exponential random variable.
► Our work partially complements earlier results by other researchers who studied convergence of extremes to a Fréchet distribution.