suppose we have $n$ Geometric$(p)$ random variables $X_1,\dots,X_n$, and let $Y_t$ be the number of these random variable still alive at time $t$, i.e.
$Y_t = \sum_{i=1}^n \mathbb{1}\{X_i \geq t\}$
I'm interested in upper bounding the expected first time that $Y_t = 0$ using a martingale approach. Here's an example argument that doesn't quite work: we can easily show that
$M_t = \log(Y_t) - \log(n(1-p)^t)$
is a supermartingale. Now we can apply the optional stopping theorem for any stopping time $\tau$ that occurs almost surely to get a bound
$\mathbb{E}[\tau] \leq \frac{1}{\log(\frac{1}{1-p})}(\log(n) - \mathbb{E}[\log(Y_{\tau})])$
However, if we let $\tau$ be the first time that all our geo random variable die, i.e. the first time that $Y_t = 0$, then we can't get anything useful from this bound, since $\mathbb{E}[\log(Y_{\tau})] = -\infty$. I'm wondering if there's a modification of this argument that works? Or if anyone knows alternate approaches? Note that this time is the same as the expected maximum of $n$ geometric random variables. But I'm specifically wondering about a martingale approach (just out of curiosity).
