# Markov Dependence in Renewal Equations and Random Sums with Heavy Tails

By Søren Asmussen and Julie Thøgersen
Thiele Research Reports
No. 02, June 2016
Abstract:

The Markov renewal equation $Z_i (x) = z_i(x) + \sum_{j \in \mathbb{E}} \int_0^x Z_j(x-y) F_{ij} (\mathrm{d} y), \qquad i \in \mathcal{E},$ is considered in the subcritical case where the matrix of total masses of the $$F_{ij}$$ has spectral radius strictly less than one, and the asymptotics of the $$Z_i(x)$$ is found in the heavy-tailed case involving a local subexponential assumption on the $$F_{ij}$$. Three cases occur according to the balance between the $$z_i(x)$$ and the tails of the $$F_{ij}$$, A crucial step in the analysis is obtaining multivariate and local versions of a lemma due to Kesten on domination of subexponential tails. These also lead to various results on tail asymptotics of sums of a random number of heavy-tailed random variables in models which are more general than in the literature.

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