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I have a math/stat problem where I need to show the convergence of the average of a sequence of experiments to an interval. The sequence of experiments is not i.i.d., hence the standard law of large ...

Let $X_i\sim N(0,\sigma_i^2)$ and $\operatorname{Corr}(X_i,X_j)>0$. Is it possible to show that $$\frac{1}{N} \sum_{i=1}^N X_i \overset{p}\rightarrow E[X_i]=0.$$ Do you have a reference to a law of ...