Eigentime identity for transient Markov chains
Let \(X_t\), \(t\geq 0\), be continuous time Markov chain with the minimal \(Q\)-function \(P(t)=(p_{ij}(t):i,j \in S)\) and \(Q\)-matrix \(Q=(q_{ij}:i,j \in S)\), the latter is assumed to be conservative, totally stable and irreducible on a countable infinite state space \(S\). Let \(\xi_0=0\), \(\xi_n=\inf\{t:t>\xi_{n-1}, X_t \neq X_{\xi_{n-1}}\}\), \(n \geq 1\), be succ\text{ess}ive jumps, \(\xi= \lim_{n \to \infty} \xi_n\) be the first leap time and \(E_i \xi\) be the expectation of \(\xi\) provided that \(X_0=i\). The proc\text{ess} is assumed to be transient, that is, the Green matrix \(G=(g_{ij}=\int_0^\infty p_{ij}(t) dt: i,j \in S)\) satisfies \(g_{ij}< \infty\) for all \(i,j \in S\). Let \(\text{tr}(G)=\sum_{i \in S} g_{ii}\) be the trace of \(G\). In the article the relationship among \(\text{tr}(G)\), \(E_i \xi\) and the eigenvalues of the corresponding Markov generator are studied. For general Markov chains it is proved that \(\sup_{i \in S} E_i \xi \leq \text{tr}(G)\). Therefore the proc\text{ess} is explosive if \(\text{tr}(G)< \infty\). For a single birth proc\text{ess} (\(S=Z_+\), \(q_{i,i+1}>0\), \(q_{i,i+j}=0\) for \(i\in Z_+\), \(j \geq 2\)) it is proved that \(\text{tr}(G)=\sup_{i \in S} E_i \xi\). Then symmetrizable proc\text{ess}es are considered, that is, \(\mu_i q_{ij}= \mu_j q_{ji}\) for all \(i,j \in S\) and for some measure \(\mu=(\mu_i>0, i \in S)\), whose total mass is infinite. Let \(L\) be the self-adjoint operator in \(L^2(\mu)\) associated with \(Q=(q_{ij})\) and let \(\sigma_{\text{ess}}(L)\) denote the \text{ess}ential spectrum of \(-L\) in \(L^2(\mu)\). For the symmetrizable proc\text{ess} with \(\sigma_{\text{ess}}(L)=\emptyset\) the general type eigentime identity \[ \sum_{n \geq 1} \lambda_n^{-1}=\text{tr}(G) \] is proved, where \(\lambda_1 \leq \lambda_2 \dots\) are all the eigenvalues of \(-L\), counting multiplicity. For example, a birth-death proc\text{ess} (\(S=Z_+\), \(q_{i,i+1}=b_i\) (\(i\geq 0\)), \(q_{i,i-1}=a_i\) (\(i\geq 1\)) and \(q_{ij}=0\) for all \(| i-j| \geq 2\)) is symmetrizable with the measure \(\mu_0=1\), \(\mu_i=b_0 b_1\dots b_{i-1}/(a_1 a_2\dots a_i)\) \((i \geq 1\)). If \(\sum_{i=0}^\infty \mu_i=\infty\) and \(R=\sum_{i=0}^\infty \mu_i \sum_{j=i}^\infty (\mu_j b_j)^{-1} < \infty\), then the eigentime formula reads \(\sum_{n \geq 1} \lambda_n^{-1}=R.\) As an application, the bounds of exponential convergence rates of (sub-) Markov semigroups \(P_t\) with the Markov generator \(L\) from \(l_\infty\) to \(l_\infty\) are given.
scientific article; zbMATH DE number 5016138