On the Harris recurrence of iterated random Lipschitz functions and related convergence rate results
In the introduction the author considers a sequence of the form \(M_n= F(\theta_n,M_{n-1})\), \(n\geq 1\), which satisfies four assumptions. \((M_n)_{n\geq 0}\) defines a temporally homogeneous Markov chain called an iterated function system (IFS) of i.i.d. Lipschitz maps hereafter. Given a Lipschitz map \(f:\chi\to\chi\), define its Lipschitz constant as \(I(f)\overset{\text{def}}= \sup_{x\neq y}{d(f(x), f(y))\over d(x,y)}\). Put \(L_n\overset{\text{def}}= I(F_n)\) for \(n\geq 1\) and note that \((L_n)_{n\geq 1}\) forms a sequence of i.i.d. random variables which is independent of \(M_0\). Its distribution does therefore not depend on the distribution of \(M_0\), that is, it is the same under which P. Elton (1990) showed that, under every \(P_x\), \((M_n)_{n\geq 0}\) converges weakly to a stationary distribution \(\pi\) provided that its Lyapunov exponent \(\ell^*\) is a.s. negative, i.e. \[ \ell^*\overset{\text{def}}= \lim n^{-1}\log\ell(F_n\circ\cdots\circ F_1)< 0\tag{1} \] and \[ \mathbb{E}\log^x L_1<\infty,\quad \mathbb{E}\log^+ d(F_1(x_0),x_0)< \infty\tag{2} \] for some \(x_0\in\chi\). He showed that \(\pi\) is unique and \((M_n)_{n\geq 0}\) is ergodic under \(P_\pi\). By Birkhoff's ergodic theorem, \[ \lim_{n\to\infty} {1\over n} \sum^n_{k=1} \mathbf{1}_B(M_k)= \pi(B)\quad P_\pi\text{-a.s.}; \] and thus also \(P_x\)-a.s. for \(\pi\)-almost all \(x\in\chi\). Hence if \(\pi(B)> 0\), then \(P_x(M_n\in B_{\text{i.o.}})= 1\) for \(\pi\)-almost all \(x\in\chi\) and it is concluded that every \(\pi\)-positive set \(B\) is recurrent. To a further analysis of IFS, the rate of convergence towards stationarity is given to the highly developed theory for irreducible Harris recurrent Markov chains on general state spaces. Given an IFS of i.i.d. Lipschitz maps satisfying (1) and (2), two questions will be considered. In Section 2 the author considers two equivalent conditions for the positive Harris recurrence on some absorbing subset \(H\) of \(\chi\) (Theorem 2.1) and provides a sufficient condition for \(H=\chi\) (Theorem 2.2). The proofs of Theorems 2.1 and 2.2 appear in Section 4. Section 3 deals with the convergence towards stationarity for Harris recurrent IFS. Under additional moment conditions on \(L_1\) and \(d(F_1(x_0), x_0)\), \(f\)-regularity and \(f\)-ergodicity for suitable functions \(f\) (Theorem 3.2) is shown. Theorem 3.1 is a consequence of results of Meyn and Tweedie (1993), the proof of Theorem 3.2 is presented in Section 5. The note contains a number of examples.
scientific article; zbMATH DE number 1901508
scientific article