I would like some help proving the following result. Thanks for any help in advance.
Let ($x_{n}), n \geq 1$ be a sequence of $\ell_{p}$ random variables for some $p$ in (1, $\infty$) and suppose that
(1) $\lim_{n\to\infty}$ $E|x_{n}|^{p} = \infty$
and
(2) $\lim_{n\to\infty}$ $P(|x_{n}| > A) = 0$ for some $0 < A < \infty$.
Prove that $\lim_{n\to\infty}$ $(E|x_{n}|)^{p} /E|x_{n}|^{p} = 0$
My initial thought was to separate out $E|x_{n}|$ into the expected value of $|x_{n}|$ multiplied by the indicator of $[|x_{n}| > A]$ plus the expected value of $|x_{n}|$ multiplied by the indicator of $[|x_{n}| \leq A]$, and then attempt to apply Holder's inequality on the former term in the sum, but I didn't get anywhere with this technique.
You have $$ \int |x_n| \; 1_{|x_n|>A} \leq \|x_n\|_p\ \ \|1_{|x_n|>A}\|_q = \|x_n\|_p \ \ P(|x_n|>A)^{1/q}$$ from which: $$ \|x_n\|_1 \leq A + \int |x_n| \; 1_{|x_n|>A} \leq A + \| x_n\|_p P(|x_n|>A)^{p/q}.$$ As $\|x_n\|_p\rightarrow \infty$ and $P(|x_n|>A)\rightarrow 0$ we conclude that $$ \frac{\|x_n\|_1}{\|x_n\|_p} \leq \frac{A}{\| x_n\|_p }+P(|x_n|>A)^{p/q}\rightarrow 0$$