Find limit of a random variable: $\lim_\limits{s\to\infty} s^n P(|Y| > s)$.

Question

$Y$ is a random variable such that ${E[|Y|^n]}<\infty$ for $n>0$.

Find $\lim_\limits{s\to\infty} s^n P(|Y| > s)$.

The answer key gives the answer $0$ without any explanation. I would assume this involves the use of Cauchy-Schwarz inequality and the fact that $E[Y^p]=\int p \, t^{p-1} \, S_x \,\mathrm{d}t$, $S_x$ being the survivor function. Any help would be appreciated thanks.

Answer 1

Write

$$s^n \mathbb{P}(|Y|>s) = \int_{\{|Y|>s\}} s^n \, d\mathbb{P}$$

to conclude that

$$s^n \mathbb{P}(|Y|>s) \leq \int_{\{|Y|>s\}} |Y|^n \, d\mathbb{P}.$$

Apply a convergence theorem of your choice to show that the right-hand side converges to $0$ as $s \to \infty$: