A Note on Comparisons of Markov Processes

Abstract

This note contains a simple proof of the following theorem of G. I. Kalmykov. Let $\{X_n\}$ and $\{Y_n\}$ be real-valued, discrete time Markov processes. Suppose $P(X_0 \leqq z) \leqq P(Y_0 \leqq z)$ for all real $z$ and $P(X_n \leqq z|X_{n-1} = x) \leqq P(Y_n \leqq z| Y_{n-1} = y)$ for $n = 1,2,\cdots$ and all $z$, whenever $y \leqq x$. Then $P(X_n \leqq z) \leqq P(Y_n \leqq z)$ for all $n$ and $z$. Some converse results are also given.