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 $Var(X) = \mathbb{E}(X^2) = \sigma^2$. I would like to bound from above the following probability as a function of $k$: $$ P\Bigl(\bigcap_{n = 1}^{k}\Bigl\{|W_n| \leq\sqrt{n}\Bigr\}\Bigr)\leq h(k) \mbox{ ?} $$ where $\lim_{k\to\infty} h(k) =0$, because I manage to prove it using zero-one Kolmogorov's law and the fact that $\frac{W_n}{\sqrt{n}}$ converges in distribution to a normal random variable.
I know that it may be a really simple question, although I just can not find the right bound.. I have tried lots of different techniques but none of them seem to work properly. I think that a useful fact is the simmetry ( since $\forall n\in\mathbb{N},W_n\sim-W_n$ ) and the fact that $W_n$ has zero mean and (in general) bounded moments.
PS: maybe a more simple question could be to bound from above $$ P\Bigl(\bigcap_{n = 1}^{k}\Bigl\{W_n \leq\sqrt{n}\Bigr\}\Bigr)\leq g(k) \mbox{ ?} $$ using a decreasing function $g(k)$ with the same properties of $h(k)$. Also note that in general: $$ P\Bigl(\bigcap_{n = 1}^{k}\Bigl\{|W_n| \leq\sqrt{n}\Bigr\}\Bigr) \leq P\Bigl(\bigcap_{n = 1}^{k}\Bigl\{W_n\leq\sqrt{n}\Bigr\}\Bigr), $$ so $g(k)$ also bounds $P\Bigl(\bigcap_{n = 1}^{k}\Bigl\{|W_n |\leq\sqrt{n}\Bigr\}\Bigr)$.
Note
The existence of such $h(k)$ has already been proven by the fact that $$ \lim_{k\to\infty} P\Bigl(\bigcap_{n = 1}^{k}\Bigl\{|W_n| \leq\sqrt{n}\Bigr\}\Bigr) = 0 $$ although I would like an "analytic" expression for it.
References
Maybe the paper "Darling-Erdős theorems for normalized sums of i.i.d. variables close to a stable law"
may help along with the question
Concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables
In fact
$$ P\Bigl(\bigcap_{n = 1}^{k}\Bigl\{W_n\leq\sqrt{n}\Bigr\}\Bigr)\leq P\Bigl(\max_{1\leq n\leq k}\frac{W_n}{\sqrt{n}} \leq 1\Bigl) $$ which could possibly be bounded from above with some decreasing function $m(k)$ using some of those results.