An inequality of the expectation

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:

$\displaystyle \begin{array}{rcl} var(Y_k) \leq \mathbb{E}[Y_k^2] & =& \int_{0}^{\infty}2y P(|Y_k|>y)dy \\ &\leq& \int_{0}^{k}2 y P(|X_1|>y)dy \end{array}$

So

$\displaystyle \begin{array}{rcl} \sum_{k=1}^{\infty} \mathbb{E}[Y_k^2]/k^2 &\leq& \sum_{k=1}^{\infty} k^{-2} \mathbb{I}_{(yy ) dy \\ &=& \int_{0}^{\infty} \left\lbrace \sum_{k=1}^{\infty} k^{-2} \mathbb{I}_{(yy ) dy \end{array}$

Now note that if ${m \geq 2}$

$\displaystyle \sum_{k \geq m} k^{-2} \leq \int_{m-2}^{\infty} x^{-2}dx = \frac{1}{m-1}$

When ${y \geq 1}$, ${k}$ takes the initial value of ${\left\lfloor y \right\rfloor +1 \geq 2}$, so

$\displaystyle 2y \sum_{k > y} k^{-2} \leq 2y / \left\lfloor y \right\rfloor \leq 4$

while when ${ 0 \leq y <1}$,

$\displaystyle 2y \sum_{k > y} k^{-2} \leq 2 \left( 1 + \sum_{k=2}^{\infty} k^{-2} \right) \leq 4$

Hence

$\displaystyle \begin{array}{rcl} \sum_{k=1}^{\infty} \frac{var(Y_k)}{k^2} &\leq& \int_{0}^{\infty} \left\lbrace \sum_{k=1}^{\infty} k^{-2} \mathbb{I}_{(yy ) dy \\ &\leq& \int_{0}^{\infty}4 P(|X_1|>y ) dy \\ &=& 4 \mathbb{E}[|X_1|] \end{array}$

$\Box$

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