A renewal theory with varying drift (Q1122877)

Let \(X_ 1,X_ 2,..\). be a sequence of independent, identically distributed random variables with mean 0 and partial sums \(S_ n=X_ 1+...+X_ n\). Put \[ T(c,\mu)=\inf \{n:\quad S_ n+n\mu >c\},\quad R(c,\mu)=S_{T(c,\mu)}+\mu T(c,\mu)-c. \] A limit r(\(\mu)\) as \(c\to \infty\) is found for the expectation of R(c,\(\mu)\) which holds uniformly over compact \(\mu\)-sets. The derivatives with respect to \(\mu\) are also considered.