Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 10F / d / Solution

For the waiting time for a new record after a fixed sample, given $U=F(Y_n)$, no overflow in the first $k$ subsequent years has probability $U^k$. Averaging as in part (c) yieldsThe tail-sum formula for expectation therefore givesThe answer is $\boxed{E\tau =\infty }$. This does not contradict almost-sure finiteness: the tail probability tends to zero, but too slowly for its sum to converge.