[Solved] Let (X_i) be independent, S_n=∑_i=1^n X_i, S_n^*=max _i ≤q n|S_i| Prove that P(S_n^*>2 a) ≤q (P(|S_n|>a))/(min _j ≤q n) P(|S_n-S_j|
Question: Let \(\left(X_{i}\right)\) be independent, \(S_{n}=\sum_{i=1}^{n} X_{i}, S_{n}^{*}=\max _{i \leq n}\left|S_{i}\right|\) Prove that
\[P\left(S_{n}^{*}>2 a\right) \leq \frac{P\left(\left|S_{n}\right|>a\right)}{\min _{j \leq n} P\left(\left|S_{n}-S_{j}\right| \leq a\right)} \quad, a>0\][Hint. If \(\left|S_{j}\right|>2 a\) and \(\left|S_{n}-S_{j}\right| \leq a\) then \(\left.\left|S_{n}\right|>a .\right]\)
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[Solved] In the setting of the previous question, prove if lim
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(See Solution) Let (X_i) be a sequence of random variables, and
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