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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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Trying to prove an inequality (looks similar to entropy)

I'm trying to prove the following inequality (or something similar, up to a constant factor in either side of the inequality): $$k\cdot\sum_{i=1}^{k}x_{i}\cdot\ln\left(x_{i}\right)\geq\sum_{i=1}^{k}x_ …