Let $S_0=0$ and $S_n = \sum_{k=1}^n Z_k$ with i.i.d real valued random variables $(Z_n)$ with $E[Z_1]=0$ and $P[Z_1 \geq 1]>0$. Let furthermore $0<\alpha<1/2$. I'm interested in lower bounds for the probabilities

$$ p_n:= P[\forall i=1,\ldots,n: S_i \geq i^\alpha] $$

for $n \rightarrow \infty$. One might try

$$ p_n \geq \prod\limits_{i=1}^n P[S_i \geq i^\alpha|S_{i-1} \geq (i-1)^\alpha] $$

and then use, if $Z_n > C$ a.s. for a constant $C > - \infty$, and suitable moment conditions hold, something like $$ P [ Si \geq i^\alpha | S_{i-1} \geq (i-1)^\alpha] \geq 1 - \frac{C}{\sqrt{i}}$$

via the Berry-Esseen theorem. However, I don't think that is optimal. I'd like to have such an inequality with $1/n$ instead of $1/\sqrt{n}$. Any suggestions?