Homogeneity and the strong Markov property (Q1819827)

This paper is devoted to the study of the relationship between homogeneity (H) and conditional independence (CI) at a stopping time \(\tau\) for a random process X. Assuming that (H) holds on the set \((X_{\tau}\in B)\), for all stopping times \(\tau\) such that \(X_{\tau}\in F\) a.s., where F is a closed recurrent subset of the state space S, while \(B\subset F\), it is shown that, when \(F=S\), (CI) holds on \((X_{\tau}\in B)\) for every stopping time \(\tau\), i.e., in this case, X is regenerative in B. If F is a proper subset of S, then the same statement is conditionally true given some shift invariant \(\sigma\)- field. Finally, the author corrects an error in his paper in Z. Wahrscheinlichkeitstheor. Verw. Geb. 60, 249-281 (1982; Zbl 0481.60019) which affects part of Theorem 4.5 there.