The first exit of almost strongly recurrent semi-Markov processes

Let ${\displaystyle \stackrel{n}{X}(·)}$, 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 ${\displaystyle \stackrel{n}{Q}}$ 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 [${\displaystyle {\pi }_j}$; j ∈ J] be the stationary p.d. of the embedded Markov chain. In terms of the averaged p.d.f. ${\displaystyle F_{\vartheta }\left(t\right):=\sum _{j,k\in J}{\pi }_j{P}_{j,k}\left(t\right)}$, t ∈ i${\displaystyle {ℝ}_{+}}$, and its Laplace-Stieltjes transform ${\displaystyle w{\tilde{F}}_{\vartheta }}$, the above assumptions imply: The time ${\displaystyle {\stackrel{n}{T}}_J}$ of the first exit of ${\displaystyle \stackrel{n}{X}(·)}$ from J has a limit p.d. (up to some constant factors) iff 1 - ${\displaystyle w{\tilde{F}}_{\vartheta }}$ is regularly varying at 0 with a positive degree, say α ∈ (0,1]. Then the transform of the limit p.d.f. equals ${\displaystyle w{\tilde{G}}^{(\alpha )}\left(s\right)={(1+s^{\alpha })}^{-1}}$, Re s ≥ 0. This extends the results by V. S. Korolyuk and A. F. Turbin (1976) obtained for α = 1 under essentially stronger conditions.