  For a right-continuous nondecreasing and unbounded function $V$ of at most
exponential growth, which vanishes on the negative halfline, we investigate the
asymptotic behavior of the Lebesgue-Stieltjes convolution powers
$V^{\ast(j)}(t)$ as both $j$ and $t$ tend to infinity. We obtain a
comprehensive asymptotic formula for $V^{\ast(j)}(t)$, 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.
