$\lim_{x \to \infty} x^p \mathbb{P}(|X| \geq x) = 0$ implies $\mathbb{E}(|X|^q)<\infty$ for $q \in (0,p)$

Question

The following problem was given in my class.

Suppose $X$ is a random variable such that $\lim_{x\to\infty} x^p P(\lvert X\rvert>x)=0$ for some $p>0$. Show that $E(\lvert X\rvert^q)< \infty$ for all $q \in (0,p)$.

I tried using indicators $I_{[X \geq x]}$ and $I_{[X

Answer 1

Hints:

1. Show that $$Y(\omega) \leq \sum_{n \geq 0} 1_{\{Y \geq n\}}(\omega)$$ for any non-negative random variable $Y$.
2. Conclude that $$\mathbb{E}(|X|^q) \leq \sum_{n \geq 0} \mathbb{P}(|X|^q \geq n). \tag{1}$$
3. Show that $$\mathbb{P}(|X| \geq n^{1/q}) \leq n^{-p/q}$$ for $n \gg 1$ sufficienlty large. Conclude that the right-hand side of $(1)$ converges for any $q \in (0,p)$.