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

Jelejko, Ya. I.; Shurenkov, V. M.

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.

Reviewer: O.K.Zakusilo (Kyïv)

MSC:

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

Keywords:

stochastic evolution; asymptotic behavior; transient phenomena