Waiting time for a new record after a fixed sample ($P(\tau =k)=\frac{n}{(n+k-1)(n+k)}$)

For independent identically distributed observations with continuous cumulative distribution function $F$, fix the maximum $Y_n$ of the first $n\ge 1$ observations and let $\tau$ be the first subsequent observation exceeding it. The probability integral transform gives $U=F(Y_n)$ density $n{u}^{n-1}$ on $(0,1)$. Conditional on $U=u$, $\tau$ has a geometric distribution of parameter $1-u$. ThereforeThe masses sum to one, so the waiting time is finite almost surely, but the tail-sum formula for expectation gives $E\tau =\infty$. These conclusions require only continuity of $F$, not a density or a strictly increasing distribution function.