Let $X_1,...X_T$ be i.i.d. random variables supported on the $[-1,1]$ segment, with an expected value of 0 and positive variance.
Let $H_t = | \sum_{i=1}^t X_i|$, and let $H = \max_{1\leq t\leq T} H_t$.
I'm interested in lower bounds on $\mathbb E[H_T]$ and $\mathbb E[H]$.
Any ideas?
Using Jensen's inequality, upper bounds are easier to get (at least for $\mathbb E[H_T]$). But what about lower bounds?
I know that if $X_i$ is Rademacher ($\pm 1$ with equal probability), then $\mathbb E[Y] = \Theta(\sqrt T)$, see here (Discrepancy between heads and tails). However, the construction in that reference uses specific properties of the Rademacher distribution, whereas here we have the general case ($X_i$ can take any bounded distribution).
I know that the law of iterated algorithm might be of use, but I have no idea how to derive a finite sample concentration.
Edit: If the distribution of $X_i$ is symmetric, we can perhaps use Khintchine's inequality. But I could not figure this out too.
Thanks!
