Consider a Probability Space $(\Omega,\mathcal{F},P)$
Let $\{X_n\}$ be a Sequence of Random variables such that $X_n \in L^2 \; \; \forall n$ and $$\sum_{j=1}^{\infty}\frac{E[X_j^2]}{j^2} < \infty $$
Consider the Natural Filtration $\mathcal{F}_n = \sigma(X_1,…,X_n)$
Assume $E[X_n|\mathcal{F}_{n-1}] = 0$
Consider $S_n = \sum_{j=1}^n X_j $ . Show that $$ \frac{S_n}{n} \xrightarrow{A.S} 0 $$
Thoughts :
If we set $Y_o = 0 $ and $Y_n = \sum_{j=1}^n \frac{X_j}{j}$
Then $\{Y_n\}$ is a Martingale with respect to the Filtration.
If we can show that $Y_n$ converges almost surely to a finite limit then Kroneckers Lemma implies the result.
My goal here is to obviously use the Martingale Convergence theorem but I'm having a small problem.
I tryed to argue that $Y_n$ is $L^1$ Bounded because $$ E[|Y_n|] \leq \sum_{j=1}^n \frac{E[|X_j|]}{j} \leq \sum_{j=1}^n \frac{E[X^2]}{j^2} $$
This would prove the result if it were true. However I only know for sure that $$ \|\frac{|X_j|}{j} \|_{L^1} \leq \|\frac{|X_j|}{j} \|_{L^2} $$
This does not neccesarily imply what I used above.
How do I fix this?
Best Answer
Since the increments of a martingale are orthogonal, the following equality holds: $$\mathbb E\left[Y_n^2\right] =\sum_{j=1} ^n\frac{1}{j^2}\mathbb E\left[X_j^2\right],$$ hence the sequence $\left(Y_n\right)_{n\geqslant 1}$ is an $\mathbb L^2$-bounded martingale.