On transience conditions for Markov chains (Q2773623)

Let \(({\mathcal X}, B)\) be a measurable space and let \(\{X_n\}_{n\geq 0}\) be an \({\mathcal X}\)-valued time-inhomogeneous Markov chain with initial value \(X_0=x\) and transition probabilities \(P(y,n,B)=\mathbf{P}\{X_{n+1}\in B\mid X_n=y\}\). Let \(L\:{\mathcal X}\to[0,\infty)\) be a measurable function and let \(\Delta_{x,m}\) denote the random variable with distribution \(\mathbf{P}\{\Delta_{x,m}\leq y\}=\text\textbf{P}\{L(X_{m+1})-L(x)\leq y \mid X_m=x\}\). Put NEWLINE\[NEWLINE \tau_{x,m}(N)=\min\{n\geq 1:L(X_{m+n})\geq N\mid X_m=x\}. NEWLINE\]NEWLINE The authors prove the following theorem. Suppose that there exist numbers \(N>0\), \(\varepsilon>0\), \(M>0\) and a measurable function \(h\:[0,\infty)\to[1,\infty)\) such thatNEWLINENEWLINE (1) \(\tau_{x,m}(N)<\infty\) a.s. for all \(x\in{\mathcal X}\) and \(m\geq 0\);NEWLINENEWLINE (2) for all \(m=0,1,2,\dots\) and all \(x\in{\mathcal X}\) such that \(L(x)\geq N\), \(\mathbf{E}\{\Delta_{x,m}I(\Delta_{x,m}\leq M)\}\geq\varepsilon\);NEWLINENEWLINE (3) the integral \(\int_1^\infty(h(t))^{-1}dt\) converges and, for \(t\geq 1\), the function \(h(t)/t\) is concave and nondecreasing;NEWLINENEWLINE (4) the family of the random variables \(\{h(\Delta_{x,m}^-);m\geq 0,L(x)\geq N\}\) is uniformly integrable.NEWLINENEWLINE Then, for all \(x\in{\mathcal X}\) and \(m\geq 0\), \(\mathbf{P}\{\lim_{n\to\infty}L(X_{m+n})=\infty \} = 1\).NEWLINENEWLINE Previously general conditions for transience have been studied by \textit{G.~Fayolle, V.~A.~Malyshev} and \textit{M.~V.~Menshikov} only in the case of countable Markov chains and under the additional assumption that the values of jumps are bounded [ see Theorem 2.2.7 in ``Topics in the constructive theory of countable Markov chains'' (1995; Zbl 0823.60053)].