(See Steps) Let (X_i) be a sequence of random variables, and let \mathcalT be its tail σ -field. Let S_n=∑_i=1^n X_i . Let b_n \uparrow ∞
Question: Let \(\left(X_{i}\right)\) be a sequence of random variables, and let \(\mathcal{T}\) be its tail \(\sigma\) -field. Let \(S_{n}=\sum_{i=1}^{n} X_{i} .\) Let \(b_{n} \uparrow \infty\) be constants. Which of the following events must be in \(\mathcal{T} ?\) Give proof or counter-example.
- \(\left\{X_{n} \rightarrow 0\right\}\)
- \(\left\{S_{n}\right.\) converges \(\}\)
- \(\left\{X_{n}>b_{n}\right.\) infinitely often \(\}\)
- \(\left\{S_{n}>b_{n}\right.\) infinitely often \(\}\)
- \(\left\{\frac{\sqrt{\sum_{i=1}^{n} X_{i}^{2}}}{S_{n}} \rightarrow 0\right\}\)
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