Convergence of Sum of Sums of random variables : trivial?

Question

I have a sequence of i.i.d random varaibles $X_1, X_2, ...$ with finite mean $\mu$ and finite variance. I also have another sequence of i.i.d random varaibles $Y_1, Y_2, ...$ with the same finite mean $\mu$ and finite variance.

I form the estimator $w_1\frac{1}{Ny}\sum_1^{Ny}{Y_i} + w_2\frac{1}{Nx}\sum_1^{Nx}{X_i}$, where $w_1 + w_2 = 1$. This is an unbiased estiamtor, however what can I say about convergence when $Nx$ and $Ny$ goes towards $\infty$? The weak and strong law of large numbers should apply to each term directly, what then happens to the convergence of their sum? Any pointer to referneces on convegence of these type of estimators is much appreciated!

Sorry if this is a tririval question...

Answer 1

Both sums converge almost surely, by the strong law of large numbers, and the fact that your variance is finite.

The sum of two convergent sequences is also convergent (fact from real analysis).

So yes, your estimator converges almost surely to $\mu$.

(Note that works even with dependence between $(X_i)_{i=1}^\infty$ and $(Y_i)_{i=1}^\infty$. In fact, if they are negatively correlated this reduces the variance of your estimate.)