I need help to prove the following equation. $X_n$ is an iid random variable, with:
$$\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=\frac{1}{2}$$
Show:
$$\mathbb{P}\left(\frac{M_n}{\sqrt{n}}>x\right)=2\mathbb{P}\left(\frac{S_n}{\sqrt{n}}>x\right)-\mathbb{P}\left(S_n=m_n\right)$$
With:
$$S_n=\sum_{i=1}^nX_i,\qquad M_n=\smash{\displaystyle\max_{j \in \{1,\dots,n\}}}S_j, \qquad m_n=\min\left\{i\in \mathbb{N}:i>x\sqrt{n}\right\}$$
Thanks for helping me with this problem. Everything is appreciated: scripts, useful tips, etc...
With best regards.