Tails of a random variable

Question

I would like to know if there are any standard tests or approaches to checking whether the following holds:

$$ \limsup_{x \rightarrow \infty} x \mathbb{P}(X \geq x) < \infty $$

where $X$ is a random variable.

Many thanks for your help.

Answer 1

If $X$ is integrable, i.e. $E(\lvert X\lvert)<\infty$, then $$x P(X\geq x)\leq xP(\lvert X\lvert\geq x)\leq \frac{xE(\lvert X\lvert)}{x}=E(\lvert X\lvert).$$

More generally, if there exists some $x_0\geq 0$ such that $\int_{x_0}^\infty t~dP(t)<\infty$, then for all $x>x_0$, $$xP(X\geq x)=\int_{x}^\infty x ~dP(t)\leq \int_x^\infty t~dP(t)\leq \int_{x_0}^\infty t~dP(t)<\infty.$$