An inequality of the mean involving truncation

Let {X_1,X_2,...} be i.i.d. r.vs with {\mathbb{E}[|X_i|] < \infty} and {Y_k = X_k \mathbb{I}_{(|X_k| \leq k)}}. Then

\displaystyle \boxed{ \mathbb{E}[X_1] \geq \sum_{k=1}^{\infty} \frac{var(Y_k)}{4 k^2 } }

Proof:

First we proove the following useful result

If {X \geq 0} and { a > 0} then

\displaystyle \boxed{ \mathbb{E}[X^a] = \int_{0}^{\infty} a x^{a-1} P(X >x) dx }

\displaystyle \begin{array}{rcl} \int_{0}^{\infty} a x^{a-1} P(X >x) dx &=& \int_{0}^{\infty} \int_{\Omega}a x^{a-1} \mathbb{I}_{(X>x)} dP dx \\ &=& \int_{\Omega} \int_{0}^{\infty} a x^{a-1} \mathbb{I}_{(X>x)} dP dx \\ &=& \int_{\Omega} \int_{0}^{X} a x^{a-1} dP dx = \mathbb{E}[X^a] \end{array}

Note you can find the same lemma on Feller Vol.2 (p. 150) as

\displaystyle \mathbb{E}[X^a]= \int_{0}^{\infty} x^a F \{ dx \} = a \int_{0}^{\infty} x^{a-1} [ 1- F(x)] dx

 

 

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Expectation: Useful properties and inequalities

Screenshot - 110614 - 16:15:09

If {X \geq 0} is a random variable on {(\Omega, \mathcal{F}, P)}. The expected value of {X } is defined as

\displaystyle \mathbb{E}(X) \equiv \int_{\Omega} X dP = \int_{\Omega} X(\omega) P (d \omega)

Inequalities

  • Jensen’s inequality. If {\varphi} is convex and {E|X|, E|\varphi(X)| < \infty}

\displaystyle \mathbb{E} (\varphi(X)) \geq \varphi(\mathbb{E}X)

  • Holder’s inequality. If {p,q \in [1, \infty]} with {1/p + 1/q =1} then

\displaystyle \mathbb{E}|XY| \leq \|X\|_p \|Y\|_q

  • Cauchy-Schwarz Inequality: For {p=q=2}

\displaystyle \mathbb{E}|XY| \leq \left( \mathbb{E}(X^2) \mathbb{E}(Y^2) \right)^{1/2}

Continue reading “Expectation: Useful properties and inequalities”