Markov dependence in renewal equations and random sums with heavy tails

Abstract

ABSTRACTThe Markov renewal equationis considered in the subcritical case where the matrix of total masses of the Fij has spectral radius strictly less than one, and the asymptotics of the Zi(x) is found in the heavy-tailed case involving a local subexponential assumption on the Fij. Three cases occur according to the balance between the zi(x) and the tails of the Fij. 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.