A contribution to large deviations for heavy-tailed random sums (Q1609657)

Let \(\{X_n:n\geq 1\}\) be a sequence of independent and nonnegative random variables with a common distribution function \(F(x)\) and \(\{N(t): t\geq 0\}\) a process of nonnegative, integer-valued random variables independent of \(\{X_n: n\geq 1\}\). \textit{C. Klüppelberg} and \textit{T. Mikosch} [J. Appl. Probab. 34, No. 2, 293-308 (1997; Zbl 0903.60021)] proved a theorem of large deviation for the random sum of \(X_n\) from 1 to \(N(t)\) when \(F(x)\) belongs to a so-called extended regular variation class and under some conditions on the convergence rate of \(N(t)\) as \(t\) tends to infinity, one of them was claimed by the authors of this paper as too strict and not even satisfied by the common renewal processes. The same result under a much weaker condition is proved in this paper and applied to a renewal risk model in insurance and finance.