Bounding the integral of the tails of a random variable.

Question

I found an argument like this in a book, but I couldn't understand how we got this bound.

Suppose $X_n$ is a sequence of random variables. For some $\delta > 0$ and all $n \geq 1$, $$ \int_{|X_n| \geq M} |X_n| \, dP \leq \frac{1}{M^\delta} E[|X_n|^{1 + \delta}]. $$

How can I justify this bound?

Answer 1

1. We don't need a sequence, only the fact that if $X$ is a non-negative random variable, then $$\int_{\{X\geqslant M\}}X\mathrm d\mathbb P\leqslant \frac 1{M^\delta}\mathbb E[X^{1+\delta}].$$

2. To see that, notice that $$X\cdot\chi_{\{X\geqslant M\}}\leqslant \frac{X^{1+\delta}}{M^\delta},$$ then integrate.