Functional limit theorems for divergent perpetuities in the contractive case*

Let ( Mk, Qk ) k∈N be independent copies of an R -valued random vector. It is known that if Yn := Q1+M1Q2+ . . .+M1 · . . . ·Mn−1Qn converges a.s. to a random variable Y , then the law of Y satisfies the stochastic fixed-point equation Y d = Q1 +M1Y , where (Q1,M1) is independent of Y . In the present paper we consider the situation when |Yn| diverges to∞ in probability because |Q1| takes large values with a high probability, whereas the multiplicative random walk with steps Mk’s tends to zero a.s. Under a regular variation assumption we show that log |Yn|, properly scaled and normalized, converge weakly in the Skorokhod space equipped with the J1-topology to an extremal process. A similar result also holds for the corresponding Markov chains. Proofs rely upon a deterministic result which establishes the J1-convergence of certain sums to a maximal function and subsequent use of the Skorokhod representation theorem.