On the approximation of the stationary distribution of a monotone stochastic Markov chain
A stochastic matrix \(P\) with states \({\mathbf N}^{++}\) is stochastically monotone if \(\sum^ m_{j = 1} p_{ij}\) is nonincreasing for \(i\) for every fixed \(m\). \(P\) is dominated by \(Q\) if \(\forall m\) \(\sum^ m_{j = 1} p_{ij} \geq \sum^ m_{j = 1} q_{ij}\). Let \(P_ n\) be \(n \times n\) matrix with \(P_ n \geq T_ n\), where \(T_ n = (p_{ij})_{i,j \leq n}\). Under the assumption that \(P\) is irreducible and dominated by an irreducible, positive-recurrent, stochastically monotone \(Q\), it is proved that if \(Q\) also satisfies the Foster-Lyapunov condition: \(\exists j_ 0, \delta\), \(0 < \delta < 1\), and \(\exists V(j)_{j \geq j_ 0}\) taking values in \([0, \infty)\), nondecreasing and \(V(j) \to \infty\), as \(j \to\infty\), such that \(\sum_{j \geq j_ 0} q_{ij} V(j) \leq (1- \delta) V(i)\), \(\forall i\geq j_ 0\), then \(\sum_{j \geq 1} | \pi_{n,j} - \pi_ j | = O\bigl({\ln V(n)\over V(n)}\bigr)\), where \(\pi_ n\) and \(\pi\) are limiting distributions of \(P_ n\) and \(P\), respectively \((\pi_{n,j} =0\) if \(j > n\)). As an application, a condition for geometric convergence is given for the random walk on the half line.
scientific article; zbMATH DE number 757554
scientific article