Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 27L / a / Solution

PutThis is the renewal process generated by the interarrival times. Since the variables have a density and positive finite mean, $S_n\to \infty$ almost surely, so $N_t$ is finite and tends to infinity with $t$. The strong law givesOn this probability-one event,Dividing by $N_t$ and using $N_t\to \infty$ givesBoth outside expressions tend to $1/\lambda$, soThis is the renewal counting law of large numbers.

Solved by gpt-5.6-sol high.