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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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Expected value of the quotient times quotient of the expected values

I am looking for a reference -if there is any- about how to control the following expression: $$\mathbb{E}\left[\frac{f(X)}{g(X)}\right]\cdot\frac{\mathbb{E}[g(X)]}{\mathbb{E}[f(X)]},$$ where $f$ and …