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Teoriya Veroyatnostei i ee Primeneniya, 2003, Volume 48, Issue 1, Pages 62–77
DOI: https://doi.org/10.4213/tvp301
(Mi tvp301)
 

Asymptotic and structural theorems for the Markov renewal equation

N. B. Engibaryan

Institute of Mathematics, National Academy of Sciences of Armenia
References:
Abstract: The multidimensional renewal equation
$$ \varphi(t)=g(t)+\int_0^t[dF(x)]\,\varphi(t-x) $$
is considered. Here $g\in L_1^n(0;\infty)$, $F(t)=(F_{ij}(t))_{i,j=1}^n$ $(n<\infty)$, $F(t)=0$ for $t\le 0$, $F(t)\uparrow$, $r(A)=1$, where $A=F(+\infty)$ and $r(A)$ is the spectral radius of the matrix $A$. For the particular case of the Markov renewal equation $\int^{n}_{i=1} F_{ij}(+\infty)=1$.
We assume that $A$ is an indecomposable matrix and a convolution power of the measure $dF$ has a nontrivial absolutely continuous component. Under these conditions it is shown that the solution of the Markov renewal equation has the form: $\varphi(t)=\mu+\rho(t)+\psi(t)$, $\rho\in C_0^n[0;\infty)$, $\psi\in L_1^n(0;\infty)$. If $dF$ is a measure with finite second moment, then $\rho\in L_1^n(0;\infty)$. Explicit formulas are obtained for $\mu$ and $\sigma=\int_0^\infty[\varphi(t)-\mu]\,dt$. Hence there follows, in particular, an asymptotic formula for $\int_0^t\varphi(x)\,dx$.
Keywords: structure and asymptotics of the solution of a multidimensional renewal equation.
English version:
Theory of Probability and its Applications, 2004, Volume 48, Issue 1, Pages 80–92
DOI: https://doi.org/10.1137/S0040585X980257
Bibliographic databases:
Language: Russian
Citation: N. B. Engibaryan, “Asymptotic and structural theorems for the Markov renewal equation”, Teor. Veroyatnost. i Primenen., 48:1 (2003), 62–77; Theory Probab. Appl., 48:1 (2004), 80–92
Citation in format AMSBIB
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\by N.~B.~Engibaryan
\paper Asymptotic and structural theorems for the Markov renewal equation
\jour Teor. Veroyatnost. i Primenen.
\yr 2003
\vol 48
\issue 1
\pages 62--77
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\crossref{https://doi.org/10.4213/tvp301}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2013405}
\zmath{https://zbmath.org/?q=an:1065.60124}
\transl
\jour Theory Probab. Appl.
\yr 2004
\vol 48
\issue 1
\pages 80--92
\crossref{https://doi.org/10.1137/S0040585X980257}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000220694300005}
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  • https://doi.org/10.4213/tvp301
  • https://www.mathnet.ru/eng/tvp/v48/i1/p62
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