Consider a random walk on a continous space, call it $W_n = \sum_{i=1}^{n}X_i$ where the $X_i$ are i.i.d. such that $\mathbb{E}(X_i) = 0$ and with finite variance. I would like to bound from below the following probability: $$ P\Bigl(\bigcup_{n =1}^{\infty}W_n\geq\sqrt{n}\Bigr)\geq \mbox{ ??} $$ It could also be written as: $$ P\Bigl(\bigcup_{n =1}^{\infty}W_n\geq\sqrt{n}\Bigr) = P(\max_{n\geq 1}\frac{W_n}{\sqrt{n}}\geq 1) $$ In particular, if I would have liked an upper bound for this quantity, then Kolmogorov's inequality would have solved my problem.
Unfortunately this is not the case and my literature search for an appropriate lower bound has failed. Thank you in advance for any help.
PS: if necessary I would add the expression of $W_n$ although I would prefer a general answer which simply use these general hypothesis.