Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 28J / d / Solution

Use the convention $J_0=0$, so $A(t)=t$ when $N_t=0$. Conditional on $N_t=n\ge 1$, the last jump $J_n$ is the largest of $n$ independent uniform points on $[0,t]$. The stated order statistic formula givesand hence, for every $n\ge 0$,Putting $\mu =\lambda t$ and averaging over $N_t$,This agrees with age of a Poisson process.