Let $\{X_n\}_{n\geq 0}$ be a random walk. Let us assume that $\mathbb{E}X_1 =0$ and $\mathbb{E}X_1^2=1$. Let also $\mathbb{E}\exp(c|X_1|)<+\infty$ for some $c>0$ and $X_1$ has a law with unbounded support. I conjecture that for any $A>0$

$\mathbb{P}(\forall_{i\in \{1,2,\ldots,n\} } X_i \geq A \sqrt{i} ) \sim n^{-C},$

where $C>0$ is some constant depending on $A$.

I can prove this claim for some special classes of RWs (e.g. with Gaussian steps). Does anyone knows general results of this kind?

Further, faster functions, e.g.

$\mathbb{P}(\forall_{i\in \{1,2,\ldots,n\} } X_i \geq \sqrt{i} \log \log i ) \sim ?,$

**Solution:**
Using the suggestions of Ofer Zeitouni (see below) I was able to make a proof of the above statement. The sketch is contained in http://goo.gl/UXfGgD.