Let $X_1, \cdots, X_n$ be $n$ random variables (not necessarily independent) such that $E[X_i] > E[X_j]$ whenever $i < j$. I am interested in obtaining lower bounds on the following probability: $P[ (X_1 > X_2) \wedge (X_1 > X_3) \wedge \cdots \wedge (X_1 > X_n) ]$.
Lower bounds on the probability that one random variable is greater than a set of others
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3Note that "lowerbounding" is not a word. There *is* a single word for this -- **minorizing** -- but it is now somewhat obscure and old-fashioned. Good mathematical prose does not call attention to itself, so I have suggested an unobtrusive alternative. – 2011-06-24
1 Answers
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The lower bound is $0$.
For example let $X_i = -(i^{k-1})$ with probability $\frac{i^2-1}{i^k-1}$ for some $k>2$, and $X_i=-\frac{1}{i}$ otherwise. Then $E[X_i]=-i$, satisfying the condition.
If $i
So $\Pr(X_1 > X_2)$ and the longer expression can be made arbitrarily small by increasing $k$.