Suppose $z_i \sim_{iid} N(0, 1)$ for all $i$ and define $$A_m = \sup_{k=1,\cdots, m}\frac{z_1^2 + \cdots + z_k^2}{k}.$$ Given $\delta$ and $m$, does anyone know the analytical solution (which may not be feasible) or a tight lower bound for the probability of $$P(A_m \leq \delta)$$, or equivalently, a tight upper bound for $$P(A_m \geq \delta).$$
If you know any papers studying this problem, please let me know.