# Expected maximum of a sequence of i.i.d. Poissons

Let $X_i \sim \mathrm{Pois}(1)$ be a sequence of $n$ i.i.d. random variables (with Poisson distribution with parameter 1). I'm interested in the asymptotic behavior of $$\mathbb E[\max_{i \in \{1\ldots n\}}X_i],$$ i.e., the expected maximum value of the sequence for large $n$.

The exact answer is $$\sum^\infty_{k = 0}\left[1-\left(\sum_{i=0}^k \frac{e^{-1}}{i!}\right)^n\right],$$ but I'm not really sure how to massage this into something that's easy to work with. I think the results I'm looking for are in a paper called "A note on Poisson maxima," but I can't find a copy of it online.

• Not a rigorous comment but here I go. Does that the expected value exist here? If you take infinitely many iid random variables with Poisson distribution, wouldn't you expect their maximum to tend to infinity also? I may be totally wrong but just wanted to share my view. – Calculon Jul 15 '14 at 19:20
• The expectation does tend to infinity, but I'm interested in its asymptotic growth. For instance, does it grow as $\Theta(\lg n)$, $\Theta(\lg \lg n)$, etc.? – Jon Jul 15 '14 at 19:25
• Just speculation, I have not calculated. Is anything useful to be obtained from hoping that the inner sum is sufficiently close to $1-\frac{e^{-1}}{(k+1)!}$? Or else use the better geometric series approximation for the tail? – André Nicolas Jul 15 '14 at 19:30
• It seems that it grows as $\log n /\log \log n$, according to arxiv.org/pdf/0903.4373.pdf – leonbloy Jul 15 '14 at 20:09
• @user159813 For an integer-valued random variable, $\sum_{t=0}^{\infty} \mathrm{Pr}(X > t)= \mathbb E X$. – Jon Jul 15 '14 at 23:10

Here is my educated guess how to tackle this problem. I would suggest to use the central limit theorem to estimate the expectation of the described above random variables and the following fact about Lp norm. $${(\sum^{N}_{i=1}x^{p}_{i}})^{1/p}\rightarrow max\{x_1,...,x_N\}$$, as p goe to infinity