Two-sided taboo limits for Markov processes and associated perfect simulation.
The joint distribution of Markov processes conditioned on staying in some fixed set is studied. Let \(X = (X_ {n},\, n\geq 0)\) be a time-homogeneous Markov chain on a Polish space \(S\) with a transition kernel \(P\). Let \(A\subseteq S\) be a Borel set, denote by \(\Gamma = \inf \{n\geq 0;\, X_ {n}\in S \setminus A\}\) the first exit time of \(X\) from \(A\). Let a sub-stochastic kernel \(K\) be obtained by restricting \(P\) to \(A\). Suppose that there exist a \(\lambda \in \left ]0,1\right ]\), a strictly positive function \(h:A\to \left ]0,\infty \right [\) and a probability measure \(\nu \) on \(A\) such that \(Kh= \lambda h\), \(K^ *\nu =\lambda \nu \) on \(A\). Set \(G(x,B) = (\lambda h(x))^ {-1}\int _ {B} h(y)K(x, dy)\) for any \(x\in A\), \(B\subseteq A\) Borel. Then \(G\) is a stochastic kernel on \(A\) and \(\eta = h\nu \) is an invariant measure for \(G\). Assume moreover that \(X\) is an aperiodic positive recurrent Harris chain on \(A\) under \(G\), hence \(\eta \) may be normalized so as \(\eta (A) = 1\). Let \(\mathbf P_ {x}\) be the probability on the path space of \(X\) under which \(X\) is a Markov chain with transition kernel \(P\) and with \(\mathbf P_ {x}\{X_ 0 =x\} = 1\), \(x\in S\). Assume that \({\widetilde{\mathbf P}}\) is the probability on the path space (of two-sided processes) under which \[ \begin{multlined} {\widetilde {\mathbf P}}(X_ {-k}\in dx_ {-k},\dots ,X_ {k}\in dx_ {k}) =\\ = \eta (dx_ {-k})G(x_ {-k},dx_ {-k+1})\cdots G(x_ {-1},dx_ 0)P(x_ 0,dx_ 1)\cdots P(x_ {k-1},dx_ {k})\end{multlined} \] for \(x_ {-k},\dots ,x_ 0 \in A\), \(x_ 1,\dots ,x_ {k}\in S\). Define \[ \mathbf P^ * (X\in \cdot ) = \left [\int \frac 1{h(X_ 0)}\, d\tilde \mathbf P\right ]^ {-1}\int \frac 1{h(X_ 0)}\mathbf 1 _ {\{X\in \cdot \}} \, d\tilde \mathbf P. \] It is shown that \(\mathbf P_ {x}(( X_ {n-k},\dots ,X_ {n+k})\in \cdot \mid \Gamma >n)\) converges in the sense of total variation to \(\mathbf P^ * ((X_ {-k},\dots ,X_ {k})\in \cdot )\) as \(n\to \infty \) for each \(k\geq 0\) fixed and for \(\nu\)-almost every \(x\). Furthermore it is proven that \(\mathbf P^ *(X_ {k+1}\in \cdot \mid X_ {j},\,j\leq k) = P(X_ {k},\cdot )\) almost surely, and many other related results are given. Moreover, analogous problems are considered for continuous-time processes and, in particular, for diffusions in \(\mathbb R^ {d}\).
scientific article; zbMATH DE number 2102357