Paper 2, Section II, F

Probability | Part IA, 2018

For a symmetric simple random walk (Xn)\left(X_{n}\right) on Z\mathbb{Z} starting at 0 , let Mn=maxinXiM_{n}=\max _{i \leqslant n} X_{i}.

(i) For m0m \geqslant 0 and xZx \in \mathbb{Z}, show that

P(Mnm,Xn=x)={P(Xn=x) if xmP(Xn=2mx) if x<m\mathbb{P}\left(M_{n} \geqslant m, X_{n}=x\right)= \begin{cases}\mathbb{P}\left(X_{n}=x\right) & \text { if } x \geqslant m \\ \mathbb{P}\left(X_{n}=2 m-x\right) & \text { if } x<m\end{cases}

(ii) For m0m \geqslant 0, show that P(Mnm)=P(Xn=m)+2x>mP(Xn=x)\mathbb{P}\left(M_{n} \geqslant m\right)=\mathbb{P}\left(X_{n}=m\right)+2 \sum_{x>m} \mathbb{P}\left(X_{n}=x\right) and that


(iii) Prove that E(Mn2)<E(Xn2)\mathbb{E}\left(M_{n}^{2}\right)<\mathbb{E}\left(X_{n}^{2}\right).

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