Let $W_n = \sum_{i = 1}^{n}X_i$ be a random walk on $\mathbb{R}$, where the increments $X_i$ are i.i.d., symmetric around the origin ($X\sim -X$), such that $-1\leq |X(\omega)| \leq 1$ $\forall\omega\in\Omega$, $\mathbb{E}(X) = 0$ and $Var(X) = \mathbb{E}(X^2) = \sigma^2 = \frac{1}{2}$.
Let $n\in\mathbb{N}$ and $I_n$ be defined as: $$ I_n = \bigcap_{k = 1}^{n} \Bigl[\frac{|W_k|}{\sqrt{k}} \leq 1\Bigr] $$
I would like to find a function $g:\mathbb{N}\to\mathbb{R}$ such that $g(n) > 0$ , $\lim_{n\to\infty} g(n) = 0$ and $$ P(I_{n-1})-P(I_n) \geq g(n) $$ for all sufficiently large $n \geq n_0$.
Of course $P(I_{n-1})-P(I_n) \geq 0$ since $I_n\subseteq I_{n-1}$, and $\lim_{n\to\infty} P(I_n) = 0$.
This question is intimately connected with this one: old question. Also, I would like to better understand the connection between random walks on the real line and the brownian motion, since the replies to the other question use a lot this idea.
The title is referring to the fact that if we're able to bound from above and from below $P(I_n)$ then we can bound from above the expression we're interesting at.
I wonder if there are some famous inequalities I miss while researching similar stuff. Thank you in advance for any hint/advice!
