Question

Suppose Xo, X1, X2, ... is a simple random walk on the integers Z withXo = 0. ) Show, for all a, b > 0 and n > 0, that

\mathbb{P}\left(\max _{k \leq n} X_{k} \geq a\right) \leq e^{-a b}\left(\frac{e^{b}+e^{-b}}{2}\right)^{n}

Fig: 1

Fig: 2

Fig: 3