Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 2 / 26J / a / Solution

For independent identically distributed positive holding times $S_i$, define $T_0=0$, $T_n={\sum }_{i=1}^n{S}_i$ and the renewal count $X_t=\max \{n:T_n\le t\}$. With $0<\mu =E{S}_1<\infty$, the renewal strong law is $X_t/t\to 1/\mu$ almost surely. The elementary renewal theorem is $m(t)/t\to 1/\mu$, where $m(t)=E{X}_t$. More generally the rate is zero when the mean holding time is infinite, under the usual nonexplosion assumptions.