Consider an infinite sequence $X = (X_i)_{i \in \mathbb N}$ of i.i.d Bernoulli random variables with (unknown) parameter $p \in (0,1)$, and let $N$ be a stopping time on $X$. Is it always the case that $\mathrm E[N]$ is a continuous function of $p$?
Intuitively this seems to be true, because the stopping time only depends on the values of $X$, and the distribution of these varies smoothly with $p$.
However, I can't find a proof or a counter-example. Any ideas?
Some approaches I have tried (unsuccessfully so far):
- Proving that the series $\sum_{n=1}^\infty n\Pr[N=n]$ converges uniformly on any interval $[a,1-a]$ with $a \in (0,1)$. This would be sufficient because $\Pr[N=n]$ is a continuous function of $p$ (in fact it's a polynomial).
- Proving that the partial sums $\sum_{n=1}^M n \Pr[N=n]$ are equicontinuous functions of $p$.
- Perhaps Wald's equality can be useful: $\mathrm E[N] \; p = \mathrm E[\sum_{i=1}^N X_i ]$.
