Exponential maximal bound for a symmetric random walk ($\mathbb{P}({\max }_{k\le n}{S}_k\ge t\sqrt{n})\le e^{-t^2/2}$)

For a simple symmetric random walk, the estimate $\mathbb{E}{e}^{\theta S_n}\le e^{n{\theta }^2/2}$ and the exponential maximal inequality give $\mathbb{P}({\max }_{k\le n}{S}_k\ge t\sqrt{n})\le e^{-t^2/2}$ for $t\ge 0$.