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?



Lower bound for small deviation probability of driftless random walk
You can find the asymptotics for this probability. 
For that let 
$$
\tau:=\inf\{i\ge 1: S\_i\le i^\alpha\}.
$$
Then, if $Var(S\_1)<\infty$ then the asymptotics for 
$$
p\_n=\mathbf P(\tau>n)\sim \frac{C}{\sqrt n},\quad n\to \infty,
$$
see <https://arxiv.org/abs/1403.5918> for random walks with i.i.d. increments, and <https://arxiv.org/abs/1611.00493> for random walks whose increments are independent but not necessarily identical. In the latter case the answer is 
$$
\mathbf P(\tau >n) \sim \frac{C}{\sqrt{Var(S\_n)}
},\quad n\to \infty. 
$$


I have doubts about existence of a simple (and accurate) lower bound. 


