An approximation theorem for Markov processes
A standard process X with state space E is considered. For every \(n\in N\) two numbers \(d_ n>h_ n>0\) and a sequence \((x^ n_ i:\) \(i\in N)\) are chosen such that \(\cup_{i}B(x^ n_ i,h_ n)=E\), where \(B(x,h)=\quad \{y:\quad d(x,y)<h\}.\) One defines \(\sigma^ n_ 1=\inf \{t: X(t)\not\in B(x^ n_ i,d_ n)\}\) on \(X_ 0\in B(x^ n_ i,h_ n)\) and \(\sigma^ n_ k\), \(k\in N\), the iterations of \(\sigma^ n_ 1.\) Let \(X^ n\) be the regular step process with parameters \(\Pi_ n(x,dy)=P^ x{\mathbb{O}}X(\sigma^ n_ 1)^{-1}\) and \(q_ n(x)=E^ x(\sigma^ n_ 1)^{-1}\). \textit{T. Watanabe} [Proc. Jap. Acad. 38, 397- 401 and 402-407 (1962; Zbl 0158.359)] has proved that the resolvents of \(X^ n\) converge to the resolvents of X. The main result of the paper is that under some supplementary assumptions on X, \(P_ n^{\mu}\to P^{\mu}\) in the Skorokhod topology. The idea of the proof is the following: if \(\tau^ n_ k\), \(k\in N\), are the jumping times of \(X^ n\), then obviously \((X(\sigma^ n_ k)\), \(k\in N)\) and \((X^ n(\tau^ n_ k)\), \(k\in N)\) are identically distributed. Then a crucial lemma expresses the following intuitive idea: If \(k_ n\) is chosen such that \(k_ nE^ x(\sigma^ n_ 1)\sim t\), then \(\sigma^ n_{k_ n}\sim t\) and \(\tau^ n_{k_ n}\sim t\).
scientific article; zbMATH DE number 3856151