Let $\{X_i \}_{i \in \mathbb{N}}$ be set of i.i.d. random variables. Define $Z_n = \sum_{i = 1}^n X_i $. I am looking for an upper bound for $\displaystyle \text{Pr} \left( Z_s \geq a \sqrt{s} \text{ for some } s \leq N \right)$.
One simple way is to use the union bound but I am looking for a tighter bound. I have looked up a number of resources for random walk (or martingale) hitting a threshold, but almost all of them have shown for a constant threshold. It seems that such a bound for time varying threshold should not be uncommon but I have not been able to figure a neat way of getting a tighter bound.
The random variables can be assumed to be subgaussian with known variance parameter. However, they need not be bounded almost surely. Also both $a$ and $N$ are assumed to be known.
Any references or hints will be appreciated. Thanks!