A property of the renewal counting process with application to the finite-time ruin probability (Q1041393)

The authors consider the renewal counting process \( \theta \left( t\right) =\sup \{n\geq 1:\theta _{1}+\cdots +\theta _{n}\leq t\},\) where \((\theta _{n})_{n\geq 1}\) is a sequence if nonnegative independent identically distributed nondegenerate random variables with finite mean. The asymptotics for the tail of the exponential moment are derived. The results obtained are applied to the finite-time ruin probability in a renewal risk model.