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Skorohod A. V. Markov processes with homogeneous second component. I.

I. I. Ezhov, A. V. Skorokhod

Kiev

Abstract: We consider Markov processes $z_t=\{x_t,y_t\}$ in a product space $X\times Y$ ($x_t\in X$, $y_t\in Y$), $Y$ being a finite-dimensional Euclidean space. Such a process is called a process with homogeneous second component if its transition probability function $P(t,x,y,s,A,B)$, $x\in X$, $y\in Y$, $A\subset X$, $B\subset Y$, $t<s$, satisfies the condition
$$ P(t,x,y,s,A,B)=P(t,x,0,s,A,B_{-y}), $$
where $B_{-y}$ is the set of $y'$'s such that $y+y'\in B$. In §1 we study general properties of such processes. In §2 the case is considered when $x_t$ is a process with denumerable set of states. §3 deals with time-homogeneous processes.

Received: 15.01.1968

 English version: Theory of Probability and its Applications, 1969, 14:1, 1–13

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