# Example of a sequence of random variables which are pairwise uncorrelated and identically distributed, but LLN does not hold

Let $$X_n$$ be a sequence of identically distributed random variables such that $$X_n$$'s are pairwise uncorrelated and $$\mathbb{E}(|X_1|)<\infty$$.

Then, is it necessary that $$\frac{X_1+...+X_n}{n} \rightarrow \mathbb{E}(X_1)$$ a.s.? What would be a counterexample?

Proof of conjecture, assuming $$E(X_1^2)$$ is finite. To simplify writing let $$Y_k=X_k-E(X_k)$$, so the problem statement is $$A_n=\frac{Y_1+Y_2+...+Y_n}{n}\to 0$$ a.s. The proof involves showing the variance of $$A_n\to 0$$, which, since $$E(A_n)=0$$, means showing $$E(A_n^2)\to 0$$.

$$E(A_n^2)=\frac{(\sum_{k=1}^nY_k)^2}{n^2}=\frac{\sum_{k=1}^nY_k^2+\sum\sum_{j\ne k}Y_jY_k}{n^2}$$ Let $$V=E(Y_1^2)$$, then $$E(A_n^2)=\frac{nV}{n^2}\to 0$$, since $$E(Y_kY_j)=0$$, for $$k\ne j$$ (uncorrelated).