$$W=\frac{\sum_{i=1}^n (X_i-\bar{X})(Y_i-\bar{Y})}{\sqrt{\sum_{i=1}^n (X_i-\bar{X})^2}\sqrt{\sum_{i=1}^n (Y_i-\bar{Y})^2}}$$
Good morning all. I am supposed to implement "the appropriate statistical theorems or methods" to answer what $W$ will converge to as the sample size $n$ approaches infinity.
At the top of my head, I saw the word converge and the sample size approaching infinity and thought of plim and the law of large numbers (LLN). I think I have to divide everything by n to obtain the sample average, and then I can say that it will converge to its expected value?
I believe the denominator will converge to the population variance (sigma squared), but what about the top?
Really hope for some insights.