Tag Archives: Inequality

Expectation: Useful properties and inequalities

If is a random variable on . The expected value of is defined as Inequalities Jensen’s inequality. If is convex and Holder’s inequality. If with then Cauchy-Schwarz Inequality: For Advertisements

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Rio’s Inequality

Let and be two integrable real-valued random variables and let be the quantile function of . Then if is integrable over we have where is the a-mixing coefficient. Proof: Set and then since and note also that which implies that

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Kolmogorov’s Maximal Inequality

Let be independent random variables with . Set . Then Proof: Let Notice that and

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An inequality of the expectation

Let be i.i.d. r.vs with and . Then Proof: So

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