Asymptotics of the generalized renewal functions when the variance is finite

We study the asymptotic behavior, as $t \to\infty$, of the generalized renewal functions
$$ \Phi_n(t)=\sum_{k=0}^\infty\frac{n\cdot(n+k-1)!}{k!}\mathsf{P}\{S_k\le t\}, $$
where $n>0$ is an integer and $S_{k}$ are partial sums of a sequence of independent identically distributed random variables with positive mean and finite variance.