Convergence of $y^q P(X>y)$ for $y \to \infty$ , $X \geq 0$ random variable, $q>0$, $\mathbb{E}[X^q]<1$

Question

Let $X \geq 0$ be a random variable, $q>0$ and $\mathbb{E}[X^q]<1$.

I want to show $y^q P(X>y) \to 0$ for $y \to \infty$.

Using indicator functions, I get $y^q P(X>y) = y^q\int_{\Omega}\mathbb{1}_{\{X>y\}}dP\leq\int_{\Omega}X^q\mathbb{1}_{\{X>y\}}dP\leq\mathbb{E}[X^q]<1$.

How can I go on from here?

Answer 1

You're on the right track using Chebyshev's inequality. The last step is to note that since $X^q$ is an $L^1$ random variable and $P(X > y) \to 0$ as $y \to \infty$, you may conclude that $$ \lim_{y \to \infty} \int_{\{X > y\}} X^q = 0 \, . $$ (e.g. by Dominated Convergence Theorem).