A sequence of real integrable ($L_1(\Omega)$) random variables $(X_n)_{n\geq 1}$ is said to satisfy the $\quad$ 1) weak $\quad$ 2) strong $\quad$ law of large numbers in case $\overline{X_n}:= \frac{1}{n} \sum_{k=1}^n (X_k - E[X_k])$ converges $\quad$ 1) in probability to 0 $\quad$ 2) almost surely to 0.
Now I found the statement that in case the sequence is pairwise uncorrelated and $M:=\sup_{k}\operatorname{Var}(X_k)<\infty $ it suffices the STRONG law of large numbers.
Using Chebyshevs inequality you find that: $P\left(|\overline{X_n}|\geq \epsilon\right) \leq \frac{M}{\epsilon^2n}$, from which one can conclude convergence in probability and so it satisfies the WEAK law of large numbers. But is there a nice argument for convergence almost surely to 0? (It was left as an exercise in my probability book, so I thought it should not be too complicated). Thank you!
Yes, there is such a classical argument, see e.g. Theorem 1.5.2 on page 21 in [1].
WLOG Var$(X_k) \le 1$, since we can divide $X_k$ by a constant. For an extension, see [2].
[1] https://yuvalperes.com/fractals-in-probability-and-analysis/ https://www.math.stonybrook.edu/~bishop/classes/math324.F15/book1Dec15.pdf
[2] Lyons, Russell. "Strong laws of large numbers for weakly correlated random variables." Michigan Math. J 35, no. 3 (1988): 353-359.