$\lim_{t \to \infty}t^p P[|X|>t]=0$ then $X \in L_{q}$

Question

I have problems concluding this exercise.

If $$\lim_{t \to \infty}t^p P[|X|>t]=0$$ for $p>0$, then $X \in L_{q}$ for all $q \in (0,p)$

I know that

$$E|X^q|=\int_{0}^{\infty}qt^{q-1}P(X>t)dt.$$

and

$$\int_{0}^{\infty}qt^{q-1}P(X>t)dt \leq \int_{0}^{\infty}pt^{p-1}P(X>t)dt.$$ Since $q \in (0,p)$, but I do not know if I could apply the limit in both sides and take it into the integral. Any help? Is this the correct approach?

Answer 1

The hypothesis $$\lim_{t \to \infty}t^p P[|X|>t]=0$$ can be relaxed to $$M:=\sup_{ 1 \le t < \infty} t^p P[|X|>t]<\infty \,.$$. Under this second hypothesis, using the formula $$E[|X|^q]=\int_{0}^{\infty} qt^{q-1}P(|X|>t) \, dt \,,$$ we get for $q<p$ that $$E[|X|^q] \le \int_{0}^{1} qt^{q-1} \, dt + \int_{1}^{\infty} qt^{q-1}Mt^{-p} \, dt =1+\frac{Mq}{p-q} <\infty \,.$$