Irreducibility results and conservativity of the absolute difference chain
Given a distribution function \(F\) on \([0,\infty)\), let \((Y_n)_{n\geq 1}\) be a sequence of i.i.d. random variables with common distribution function \(F\). Define recursively \(X_{n+1}=| X_n- Y_{n+1}|\), \(n\geq 0\), where \(X_0\) is a given random variable independent of \((Y_n)_{n\geq 1}\). Then the sequence \((X_n)_{n\geq 0}\) is a Markov chain called the \((F\)-)absolute difference chain started at \(X_0\). First considered by W. Feller, such chains were subsequently studied by \textit{P. S. Puri} and \textit{H. Rubin} [Ann. Math. Stat. 41, 2113--2122 (1970; Zbl 0225.60007)] and \textit{F. B. Knight} [Z. Wahrscheinlichkeitstheorie Verw. Gebiete 43, 57--63 (1978; Zbl 0383.60060)]. The present author gives a sufficient condition (derived jointly with S. N. Smirnov in 1987) and a necessary one for the \(\lambda\)-irreducibility of the chain \((X_n)_{n\geq 0}\). Here, \(\lambda\) is Lebesgue measure on \([0,\infty)\). It is also shown that if \(EY^{1/2}_1< \infty\), then \((X_n)_{n\geq 0}\) is conservative [cf., e.g., \textit{S. R. Foguel}, ``The ergodic theory of Markov processes'' (1969; Zbl 0282.60037), Chapter 2].
scientific article; zbMATH DE number 2052900