Convergence in $L^p$: Marcinkiewicz-Zygmund Strong Law of Large Numbers

Question

If $(X_n)_n$ is a sequence of random variables i.i.d, let $0<p<2,$ $Y_n=\sum_{k=1}^nX_k.$

We can prove, using Kolmogorov three series theorem, Marcinkiewicz-Zygmund Strong Law of Large Numbers:

If $X_1 \in L^p$ then $\lim_n\frac{Y_n}{n^{1/p}}=0 \ a.s$ if $p<1$ and $\lim_n\frac{Y_n-nE[X_1]}{n^{1/p}}=0 \ a.s$ if $1\leq p<2$.

I would like to know if there is convergence in $L^p.$

Answer 1

Let $0<p<2$. Suppose that $X, X_1, X_2, \cdots$ are i.i.d. random variables and $ S_n=\sum\limits_{k=1}^{n}X_k, n\ge 1 $. If $\mathsf{E}|X|^p<\infty$, and $\mathsf{E}X=0$ when $1\le p<2$, then $$ \lim_{n\to\infty}\mathsf{E}\Big|\frac{S_n}{n^{1/p}}\Big|^p= \lim_{n\to\infty}\mathsf{E}\frac{|S_n|^p}{n}=0. $$

You could find the proof of above conclusion in the book: A. Gut, Probability: A Graduate Course, 2nd ed., Springer Verlag, 2013. Th 10.3, p.311.