Let $X_i$ be a sequence of i.i.d. standard normal random variables. Let $Y_i=\sum_{k=1}^iX_k$ and $Z_i=\sum_{k=1}^iY_k$. I am interested in upper and lower bounds for $P(\sup_{1\leq i\leq m}|X_i|\leq c)$, $P(\sup_{1\leq i\leq m}|Y_i|\leq c)$ and $P(\sup_{1\leq i\leq m}|Z_i|\leq c)$. I managed to figure out the first two, and for the second one I got bounds of something like $1-\mbox{const}\times e^{-c^2/n}$ by considering $P(|B_t|\geq c)$ and using the reflection principle for $\tau=\inf_{t\leq m}\{t: \ |B_t|\geq c)\}$.
The problem is, the same trick doesn't quite work when figuring out the $P(\sup_{1\leq i\leq m}|Z_i|\leq c)$. In fact, the best I could do was write $Z_{n+1}=Z_n+B_{n+1}$ where $B$ is a standard Brownian motion. Maybe this can become a stochastic differential equation? But, it feels intractable unless I'm missing something. I was wondering how one might get good upper and lower bounds for the supremum over $Z_i$? Maybe I'm thinking too hard and there's an easier way which doesn't resort to Brownian motion.
Any help would be greatly appreciated!