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 exten...