Let $X, X_i$, $1 \leq i \leq n$ be IID random variables such that:
$$\mathbb{E}X^2 = +\infty$$I am trying to show that this implies a growth condition on$$\mathbb{E}[\max_{1\leq i \leq n}{|X_i|}]$$,Specifically, I want to show for any $\epsilon > 0$ there exists some $c = c(\epsilon)$ such that
$$\mathbb{E}[\max_{1 \leq i \leq n}|X_i|] \geq c n^{1/(2+\epsilon)}$$.
Our instructor suggested that we prove the statement using contraposition, e.g. we show that:$$\liminf \frac{\mathbb{E}[\max_{1 \leq i \leq n}|X_i|]}{n^{1/(2+\epsilon)}} = 0 \implies \mathbb{E}[X^2] < +\infty $$My attempt:The above condition is equivalent to saying that there exists a subsequence $n_k$ that:$$\lim_{k \to \infty} \frac{\mathbb{E}[\max_{1 \leq i \leq n_k}|X_i|]}{n_k^{\frac{1}{2+\epsilon}}} = 0$$Let $a_n$ be any monotone increasing sequence of positive real numbers such that $\lim a_n = +\infty$. By the monotone convergence theorem and Fubini's theorem:$$\mathbb{E}[X^2] = \int_0^\infty \mathbb{P}(X^2 > t) dt = \sum_{n=1}^\infty \int_{a_n}^{a_{n+1}}\mathbb{P}(X^2 > t) dt \leq \sum_{n=1}^\infty \underbrace{(a_{n+1} - a_n)}_{:= \Delta a_n} \mathbb{P}(X^2 > a_n) $$Thus, it suffices to show that the sum on the right hand side is finite for $a_n$ appropriately chosen.
To use information about the maximum, we have for $t> 0$ and $M_n = \max_{1\leq i \leq n} |X_i|$ the running maximum that:$$\mathbb{P}(M_n >t) = 1 - (\mathbb{P}(X\leq t))^n = 1 - (1 - \mathbb{P}(X > t))^n$$Rearranging gives:$$\mathbb{P}(X > t) = 1 - (1 - \underbrace{\mathbb{P}(M_n > t)}_{x})^{1/n}$$The right hand side is monotone as a function of the argument $x$. Hence, applying Markov's inequality to $M_n$ gives:$$\mathbb{P}(X > t) \leq 1 - \left(1 - \frac{\mathbb{E}[M_n]}{t}\right)^{1/n}$$for any integer $n$ and nonnegative number $t$. Replacing $X$ with $X^2$ gives:
$$\mathbb{P}(X^2 > t) = \mathbb{P}(X > t^{1/2}) \leq 1 - \left(1 - \frac{\mathbb{E}[M_n]}{t^{1/2}}\right)^{1/n}$$Hence, for $a_k$ as above, we have:$$\mathbb{E}[X^2] \leq \sum_{k=1}^{\infty} \left(1 - \left(1 - \frac{\mathbb{E}[M_n]}{a_k^{1/2}}\right)^{1/n}\right)\Delta a_k$$
My question now is - what is the right choice of sequence $a_n$? I was considering using the subsequence along which the quotient:
$$\mathbb{E}[M_{n_k}]/{n_k}^{1/(2+\epsilon)} \to 0 $$
But this scares me a bit because I can't control the size of the increments $\delta n_k$ (the subsequence could be very sparse). One way I was thinking of dealing with that was through summation by parts, but I'm not sure if that would do anything.
Any hints or ideas would be appreciated!