Asymptotic and structural theorems for the Markov renewal equation

N. B. Engibaryan

Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 1, pp. 62-77

Résumé

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 $\int_0^t\varphi(x)\,dx$.

Détail

Keywords: structure and asymptotics of the solution of a multidimensional renewal equation.