Asymptotically linear iterated function systems on the real line

Abstract

Given a sequence of i.i.d. random functions Ψ n : R → R, n ∈ N, we consider the iterated function system and Markov chain which is recursively defined by X x 0 := x and X x n := Ψ n−1 (X x n−1 ) for x ∈ R and n ∈ N. Under the two basic assumptions that the Ψ n are a.s. continuous at any point in R and asymptotically linear at the "endpoints" ±∞, we study the tail behavior of the stationary laws of such Markov chains by means of Markov renewal theory. Our approach provides an extension of Goldie's implicit renewal theory [20] and can also be viewed as an adaptation of Kesten's work on products of random matrices [24] to one-dimensional function systems as described. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics, e.g. ARCH models and random logistic transforms.

On uniqueness of invariant measures for random walks on

“…These are sufficient conditions for (24) but examples of systems which do not satisfy these conditions, but for which (24) still holds, might be easily provided. Processes of this kind appear in many contexts of probability and related fields and have been investigated in several paper in the last years; see, for example, [2,3,7,8,14,20]. A fundamental example that has been widely studied is the affine recursion where g(x) = A(g)x + B(g) (see [9] for a general overview).…”