A Note on the Convergence of Probability Functions of Markov Chains

#### J. Baczynski, M.D. Fragoso

2013, v.19, Issue 1, 141-148

ABSTRACT

Let $P(t)\equiv(p_{ij}(t))_{i,j\in \mathbb{N}}$ stands for a generalized transition matrix function of a continuous-time homogeneous Markov chain with an infinite countable state space and denote $o_{ij} (h) := p_{ij}(h) - p_{ij}(0) - \dot{p}_{ij}(0)h$. In this note we are given conditions to have the convergence $o_{ij} (h)/ h \rightarrow 0$ as $h\rightarrow 0$ performing uniformly with respect to $i$ for each $j$. We also provide some examples where this convergence do not hold. In addition, we identify a liaison between these issues and the uniqueness of solutions of the Kolmogorov forward equation.

Keywords: Markov chains,Kolmogorov forward equation,probability transition matrix function