Convergence of $\sum_{i \leq n} X_i/n$ I have a question like this:
Let $(X_n)$ be an i.i.d sequence of random variables with values in $\{-1,1\}$, and define $Y_n:= \sum_{i \leq n} X_i/n$. Show that $(Y_n)$ converges almost surely and in $L_1$. Do not appeal to the law of large numbers.
The question is under the classification of martingale. I know we should be able to do this using backwards martingales. but we never learned the backwards martingales in class and can we do it using regular martingales? I mean, can we define a suitable filtration so that the $(Y_n)$ is indeed a martingale?

I do not know how one can use the particular Rademacher distribution, but there is a useful reverse martingale trick: it is easy to see that 
$$
E[X_1|S_n, S_{n+1},\dots] = \frac{S_n}{n}.
$$
Denoting $\mathcal{F_n} = \sigma\{S_n, S_{n+1},\dots\}$, we thus have a (reverse) Levy martingale $Y_n = E[X_1|\mathcal{F_n}]$. By the reverse martingale convergence, $Y_n\to Y_\infty = E[X_1|\mathcal{F_\infty}]$, $n\to\infty$, where $\mathcal{F}_\infty = \cap_{n\ge 1} \mathcal F_n$. But $\mathcal{F}_\infty$ consists of exchangeable events, which by the Hewitt-Savage 0-1 law have probability $0$ and $1$. Therefore, $Y_\infty = E[X_1]$ a.s.
