Do you know any kind of generalisation of the law of large numbers. I mean something like this :
Assume that $(X_n)_{n\in\mathbb{N}}$ is a sequence of independant variables (not necessarily identically distributed), which have all the same mean $\mathbb{E}(X_1)$. Then, $$ \frac{X_1 + X_2 + \dots + X_n}{n} \xrightarrow[n\to\infty]{a.s.} \mathbb{E}(X_1)$$
Or, more precisely, in the case I am considering, something like this :
Assume that $(X_n)_{n\in\mathbb{N}}$ is a sequence of independant Gaussian variables, which have not all the same variance but which have all the same mean $\mathbb{E}(X_1) = 0$. Then, $$ \frac{X_1 + X_2 + \dots + X_n}{n} \xrightarrow[n\to\infty]{a.s.} \mathbb{E}(X_1) = 0$$
Thank you for your consideration on this matter,
Nawak
The statement for the independent non-identically distributed random variables do not hold in general. Indeed, it would imply that $X_n/n\to 0$ almost surely and by the Borel-Cantelli lemma we should have $\sum_{n=1}^{+\infty}\mathbb P\{X_n\geqslant n\}<+\infty$. In order to construct a counter-example, we can take $X_n$ taking the values $n$, $-n$ and $0$ with respective probabilities $1/(2n)$, $1/(2n)$ and $1-1/n$.
The second statement holds if $\sum_{i=1}^{+\infty}\mathbb{E}\left[X_i^2\right]/i^2<+\infty$. This can be seen from the martingale convergence theorem applied to the martingale $\left(\sum_{i=1}^nX_i/i, \sigma\left(X_i,1\leqslant i\leqslant n\right)\right)_{n\geqslant 1}$.