ε-REGENERATIVE PHENOMENA, p-α PAIRS AND MARKOV TRANSITION PROBABILITIES

Suppose that for each positive numbert, E（t）andA（t）are disjoint events. Let J₁（t）,J₂（t）andJ_ε（t）denote E（t）,A（t） and E（t）∪A（t）.respectively LetJ（t,L）denote\bigcup_l \in L J_l(t),whereL∈ 1, 2, 3. If for any 00<t₁﹤…<t_i+g, we have P(J（t₁,L₁）…J(t_i-1,L_i-1)E（t_i）...J(t_i+1,L_i+1)…J(t _i+g,L_i+g)=P(J（t₁,L₁）…J(t_i-1,L_i-1)E（t_i）)P(J(t_i+1一t_i,L_i+1)…J(t_i+g一t_i,L_i+g),then(E（t）,A（t):t＞0will be called ε-regenerative phenomenon and (p（t）,a（t）) the correspondingp-a pair, wherep（t）=P(E（t）),a（t）=P(A（t）) Let \lim _t \rightarrow 0 p(t)=1. Then (p（t）,a（t）) is a p-apair if and only if there are Markovtransition functions P_t（·,·）, standard state x, measurable set B, x∈B, such thatp（t）=P_t,（x,x）,a（t=P_t（x,B）; if and only if a（t） is continuous, p（t） is a p-function （with canonical measure μ）1, and there is a measurable function g（s） such that 0≤g（s）≤μ(s, ∞ and a(t)=\int_0^t p(t-s) g(s) d s. The limits and products of p-a pairs are also p-a pairs. Some conditions for a p-a pair to be finite decomposable and to be indecompo-sable are given.