Let $\stackrelnX(·)$, n ∈ N, be a sequence of homogeneous semi-Markov processes (HSMP) on a countable set K, all with the same initial p.d. concentrated on a non-empty proper subset J. The subrenewal kernels which are restrictions of the corresponding renewal kernels $\stackrelnQ$ on K×K to J×J are assumed to be suitably convergent to a renewal kernel P (on J×J). The HSMP on J corresponding to P is assumed to be strongly recurrent. Let [$π_j$; j ∈ J] be the stationary p.d. of the embedded Markov chain. In terms of the averaged p.d.f. $F_{ϑ}(t) :=\sum_{j,k ∈ J} π_jP_{j,k}(t)$, t ∈ i$ℝ_+$, and its Laplace-Stieltjes transform $\widetilde F_ϑ$, the above assumptions imply: The time $\stackrel{n}{T}_{J}$ of the first exit of $\stackrel{n}{X}(·)$ from J has a limit p.d. (up to some constant factors) iff 1 - $\widetilde F_ϑ$ is regularly varying at 0 with a positive degree, say α ∈ (0,1]. Then the transform of the limit p.d.f. equals $\widetilde G^{(α)}(s) = (1+s^{α})^{-1}$, Re s ≥ 0. This extends the results by V. S. Korolyuk and A. F. Turbin (1976) obtained for α = 1 under essentially stronger conditions.
We define and give the various characterizations of a new subclass of geometrically infinitely divisible random variables. This subclass, called geometrically semistable, is given as the set of all these random variables which are the limits in distribution of geometric, weighted and shifted random sums. Introduced class is the extension of, considered until now, classes of geometrically stable [5] and geometrically strictly semistable random variables [10]. All the results can be straightforward transfered to the case of random vectors in $ℝ^d$.
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.