题号:14636    题型:解答题    来源:2024年丘成桐大学生数学竞赛(概率与统计)-无答案
Let $X_1, X_2, \cdots$ be independent Bernoulli random variables $s$ atisfying $P\left(X_i=1\right)=p$ and $P\left(X_i=-1\right)=q=1-p$ for some $p \in(0,1)$. Let $S_n=X_1+\cdots+X_n$ and $M=\sup _{n \geq 1}\left(S_n / n\right)$.
(a) Calculate $P(M=0)$.
(b) Show that $P(p-q < M \leq 1)=1$. For any rational num ber $x \in(p-q, 1]$, is $P(M=x)>0$ ? If so, prove it. If not, fi nd a point with zero probability.
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