Suppose $X$ is a random variable, $a>0$ is a constant. Prove that $E\vert X \vert^{a}<\infty$ iff $\sum\limits_{n=1}^{\infty}n^{a-1}P(\vert X \vert \geqslant n)<\infty$.
Prove that $E\vert X \vert^{a}<\infty$ iff $\sum\limits_{n=1}^{\infty}n^{a-1}P(\vert X \vert \geqslant n)<\infty$
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