It's a problem from 'High Dimensional Statistics' (MIT's lecture notes)
(Problem 1.3,page30)Let $X_1,X_2$ ... be an infinite sequence of sub-Guassian random variables with variance proxy $\sigma_{i}^{2}=C(\log i)^{-1/2}$. Show that for C large enough, we get:
$$ E[\max_{i\geq2}X_i]< \infty $$ QU:
I find it a bit counter-intuition because I feel when C is larger, the expectation will be larger and more likely to get $\infty$ since the tail is thicker. But if I didn't misunderstand it, the problem says 'We need a C large enough to get a finite expectation'. Or it just says the expectation will be finite no matter how large the C is? Under both interpretations, I fail to prove it. :(
Ps: Definition: A random variable X is said to be sub-Guassian with variance proxy $\sigma^2$ if $E[X]=0$ and its moment generating function satisfies: $$ E[\exp(sX)]\leq \exp(\dfrac{\sigma^2s^2}{2}), \forall s\in R $$ It's just an extension to guassian's.