Asymptotically linear iterated function systems on the real line

Given a sequence of i.i.d. random functions Ψn:R→R, n∈N, we consider the iterated function system and Markov chain, which is recursively defined by X0x:=x Xnx:=Ψn−1(Xn−1x) for x∈R n∈N. Under two basic assumptions that Ψn are a.s. continuous at any point in R asymptotically linear “endpoints” ±∞, study tail behavior stationary laws such chains means renewal theory. Our approach provides an extension Goldie’s implicit theory (Ann. Appl. Probab. (1991) 1 126–166) can also be viewed as adaptation Kesten’s work on products matrices (Acta Math. (1973) 131 207–248) to one-dimensional systems described. results have applications quite different areas applied probability like queuing theory, econometrics, mathematical finance population dynamics, example, ARCH models logistic transforms.