0
$\begingroup$

Let $X_1,\ldots,X_n\sim \mathit{Geo}(p)$ be independent random variables, and let $M=\max\{X_1,\ldots,X_n\}$ denote their maximum.

Given a parameter $\delta\in(0,1)$, I'm looking for a bound $T(n,p,\delta)$ of the form $$ \Pr[M > T(n,p,\delta)]\le \delta. $$


A simple solution can be derived using the union bound. If we demand $$ (1-p)^T = \Pr [X_i> T] \le \delta/n, $$ we have $T=\frac{\ln(\delta/n)}{\ln(1-p)}$, and a union bound over all $X_i$'s show that this holds.

However, I think that this is a rather loose bound, and some simulations I did seem to agree.

  • How can we get a tighter bound (ideally, a close-form bound that is possible to work with)?
$\endgroup$
3
$\begingroup$

$M=\underset{1\le i\le n}{\max} X_i.$ This means $M<T\,\,$ iff $\,\, X_i< T\,\,\forall \,\,i.$ Hence $\mathbb{P}(M<T)=\left[\mathbb{P}(X_i<T)\right]^n=\left[1-(1-p)^T\right]^n.$

$$\text{You need }\left[1-(1-p)^T\right]^n=1-\delta$$ $$\iff 1-(1-\delta)^{1/n}=(1-p)^T\iff \,\,T=\dfrac{\log\left(1-(1-\delta)^{1/n}\right)}{\log(1-p)}.$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.