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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

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upper bound of the expectation of upper quantile ratio

Your statement is true. This inequality is equivalent to the fact that the correlation of $\frac{\sum S_{\geq p}}{\sum S}$ and $\sum S$ is positive correlated. More precisely: For a given $q$, let $ …
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