Recently I have considered the following question: for a sequence of i.i.d. random variables(maybe normal distribution), the expectation of the maximum of n such kind of variables is dependent on n, so if n goes to infinity, will the expectation of the maximum of infinite i.i.d random variables be infinite? What is the upper bound of it?
[Math] expectation of maximum of infinte iid random variables
probability
Best Answer
If the support of your distribution is not bounded above (e.g., for a Gaussian distribution), then you will have for every $x$ that there is some $\epsilon > 0$ so that $P(X_j > x) = \epsilon$ for all your i.i.d. variables $X_j$. So the probability that no $X_j$ is greater than $x$ is $(1 - \epsilon)^n$ if you have $n$ variables, which goes to $0$ as $n \to \infty$. Thus by the Markov inequality, your expectation for the maximum must go to $\infty$ as $n \to \infty$.