## Transient phenomena in a class of matrix-valued stochastic evolutions.(English. Ukrainian original)Zbl 0943.60053

Theory Probab. Math. Stat. 52, 75-79 (1996); translation from Teor. Jmovirn. Mat. Stat. 52, 72-76 (1995).
Let $$x(t)$$ be a regenerative process with the regeneration moments $$\tau_1 =\tau, \tau_2,\tau_3,\dots,$$ where $$\tau_1,\tau_2 -\tau_1,\dots,\tau_n -\tau_{n-1},\dots$$ are independent identically distributed random variables with non-arithmetic distribution and finite expectation $$E\tau =\mu <\infty.$$ Consider a matrix-valued stochastic evolution given by the equation $$dT^{\varepsilon}(t)/dt =T^{\varepsilon}(t) A^{\varepsilon}(x(t))$$, $$T^{\varepsilon}(0)=I.$$ Assume that $$A^{\varepsilon}(x)$$ is a matrix-valued measurable function with non-negative off-diagonal elements and $$\|A^{\varepsilon}(x)\|\to 0$$ as $$\varepsilon \to 0$$ uniformly in $$x.$$ Theorem 1 asserts that the condition $$E T^{\varepsilon}(\tau)=I+a(\varepsilon) C+o(a(\varepsilon))$$, $$a(\varepsilon)\to 0$$, implies $$\lim_{\varepsilon\to 0} E T^{\varepsilon} (t/a(\varepsilon))=e^{tC/\mu}.$$ The asymptotic behavior of $$E T^{\varepsilon}(\tau)-I$$ is investigated.

### MSC:

 60H25 Random operators and equations (aspects of stochastic analysis) 60K15 Markov renewal processes, semi-Markov processes