Convolution powers of unbounded measures on the positive half-line

Abstract

For a right-continuous nondecreasing and unbounded function V of at most exponential growth, which vanishes on the negative half-line, we investigate the asymptotic behavior of the Lebesgue-Stieltjes convolution powers $V^{⁎\left(j\right)}\left(t\right)$ as both j and t tend to infinity. We obtain a comprehensive asymptotic formula for $V^{⁎\left(j\right)}\left(t\right)$, which is valid across different regimes of simultaneous growth of j and t. Our main technical tool is an exponential change of measure, which is a standard technique in the large deviations theory. Various applications of our result are given.