Let $(X_i)$ be a sequence of stationary random variables. Why is the inequality $$ \sum_{i < j}^{n} \mathbb{P}[X_i > \alpha, X_{j} > \alpha] \leq n \sum_{j=2}^{n} \mathbb{P}[X_{1} > \alpha, X_{j} > \alpha] $$ correct?
2026-03-25 16:08:28.1774454908
Why is $\sum_{i<j}^n \mathbb{P}[X_i > \alpha, X_j > \alpha] \leq n \sum_{j=2}^n \mathbb{P}[X_1 > \alpha, X_j > \alpha]$?
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Hints: write LHS as $\sum_i \sum_{j=i+1}^{n} P\{X_1 >\alpha, X_{j-i+1} >\alpha \}$ where I have used the fact that $(X_i,X_j)$ has the same joint distribution as $(X_1,X_{(j-i+1)})$. Now just interchange the sums on the left side and you will be able to get the stated inequality.