Abstract
We consider the renewal counting process \( \mathit{\Theta} \left( t \right) = \sup \left\{ {n \geqslant 1:\theta_1 + \cdots + \theta_n \leqslant t} \right\} \), where θ1, θ2,… are nonnegative independent identically distributed nondegenerate random variables with finite mean. The asymptotics for the tail of the exponential moment are derived. The obtained results are applied to the finite-time ruin probability in a renewal risk model.
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