Functional limit theorems for divergent perpetuities in the contractive case. (English) Zbl 1307.60026

Buraczewski, Dariusz; Iksanov, Alexander

Electron. Commun. Probab. 20, Paper No. 10, 14 p. (2015).

Summary: Let \((M_k, Q_k)_{k\in \mathbb{N}}\) be independent copies of an \(\mathbb{R}^2\)-valued random vector. It is known that if \(Y_n:=Q_1+M_1Q_2+\ldots+M_1\cdot\ldots\cdot M_{n-1}Q_n\) converges a.s. to a random variable \(Y\), then the law of \(Y\) satisfies the stochastic fixed-point equation \(Y =^d Q_1+M_1Y\), where \((Q_1, M_1)\) is independent of \(Y\). In the present paper we consider the situation when \(|Y_n|\) diverges to \(\infty\) in probability because \(|Q_1|\) takes large values with high probability, whereas the multiplicative random walk with steps \(M_k\)’s tends to zero a.s. Under a regular variation assumption we show that \(\log |Y_n|\), properly scaled and normalized, converges weakly in the Skorokhod space equipped with the \(J_1\)-topology to an extremal process. A similar result also holds for the corresponding Markov chains. The proofs rely upon a deterministic result which establishes the \(J_1\)-convergence of certain sums to a maximal function and subsequent use of the Skorokhod representation theorem.

MSC:

60F17	Functional limit theorems; invariance principles
60G50	Sums of independent random variables; random walks
60G70	Extreme value theory; extremal stochastic processes

Keywords:

functional limit theorem; perpetuity; random difference equation; extremal process