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12] where U (x, [0, t]×A) = Px (τn ≤ t, Xτn ∈ A). For ſxed x this kernel is a mea∞ sure of intensity of some random point ſeld. So, let N (B×A) = n=0 IB×A (τn , Xn ) be a number of pairs (time, point) of discontinuity belonging to a set B × A (B ∈ B(R+ ), A ∈ B(X)). Then ∞ Ex N (B × A) = Ex IB×A τn , Xn n=0 ∞ Px τn ∈ B, Xτn ∈ A = U (x, B × A). = n=0 19. e. for renewal processes. According to the common scheme the renewal process is a Markov renewal process with space of states X = N0 and with deterministic passages {Xn = n} → {Xn+1 = n + 1}.

Where Bn = f −1 ατ−1 n From here, using the condition of regeneration of the family (Qx ), we obtain EQx QX n f −1 B ; Bn = EQx PX n (B); Bn = EQx PXτn ◦f (B); Bn C ∩ τn < ∞ . 11], is called the semi-Markov (stepped) family of measures. The random process, being determined within the initial point by this family of measures, is said to be a stepped semi-Markov process. For the semi-Markov family of measures (Px ) the kernel F ([0, t] × A | x) means a conditional distribution of a pair (time, point) of the ſrst jump: Px τ1 ≤ t, Xτ1 ∈ A = Qx X1 ≤ t, X 1 ∈ A = F [0, t] × A | x .

In other words such a process has a Markov property with respect to the ſrst exit time from any open set. Problems of regeneration times have been investigated by many authors (see Shurenkov [SHU 77], Mainsonneuve [MAI 71, MAI 74], Mürmann [MUR 73], Smith [SMI 55], Taksar [TAK 80]). In this chapter we research a structure of classes of regeneration times connected with families of probability measures depending on a parameter x ∈ X. The closure of such a class with respect to some natural compositions of Markov times makes it possible to be limited by a rather small set of these times when deſning a semi-Markov process for its constructive exposition.

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