Let $X_i$ be an i.i.d sequence of symmetric real random variables with variance 1 and $S_n = X_1 + \dots + X_n$. Let $Z_n = \max_{1 \leq k \leq n } (|S_k|)$.
$Z_n$ is typically of the order of $C \sqrt{n}$. I am interested in the probability that $Z_n$ is very small compared to $\sqrt{n}$.
More precisely
let $a_n$ be a sequence of positive real numbers diverging to $\infty$ such that $a_n/\sqrt{n} \to 0$. I am specifically interested in the case where $a_n = n^{1/3}$.
I am hoping of a result like $\frac{a_n^2}{n} \log P(Z_n \leq a_n ) $ converges to some known constant.
How can I get such a result?
Here are something I know but that are insufficient:
This can be computed for the maximal absolute value of a standard Brownian motion when time goes to infinity as the distribution can be computed. but I am unable to find something similar for random walks or to use the Brownian limit by using explicit distribution of maximum which can be found in this post.
There exists positive constants $C,K,k, c$, such that for all $M \geq 1 $ and for $N$ large enough:
$ K\exp(-C N/M^2)\leq P(Z_N \leq a) \leq k \exp(-c N/M^2)$ which gives some reasonable hope that whar I am asking may be true. This can be found in Potential theory and geometry on Lie groups by Varopoulos (chapter 2 Annex and chapter 5 (§5.12.1) ).
- I have tried to see if large deviation theory could help to prove this but I could not find anything and I don't know much about this topic, but this looks a bit like Cramér's theorem.
I am asking this question because I am reading Varopoulos book and I am trying to better understand the behavior of the random walk on certain amenable Lie groups but I don't think the context is very relevant to my question in this particular case.
