What will U converge to? 
Hi all,
I have several queries below:

*

*Would the X-bar and Y-bar be fixed in value?


*Would the denominators become exceedingly large as n increases?


*Would the numerator become exceedingly large as well as n increases?


*In that case, wouldn't the value of U converge towards 1?  (As both numerator and denominator become very large)
Many thanks in advance and cheers!
 A: For the first part, you have paired data $(X_i,Y_i), i=1,2,\dots,6.$
x = c(74, 61, 63, 85, 57, 76)
y = c(1.71, 1.68, 1.62, 1.78, 1.56, 1.73)

Then the sample covariance is $S_{xy} = \frac{\sum_{i=1}^6(X_i-\bar X)(Y_i-\bar Y)}{n-1}.$
The sample standard deviation of the $X_i$s is
$S_x = \sqrt{\frac{\sum_{i=1}^n(X_i - \bar X)^2}{n-1}}$
and similarly for the sample SD of the $Y_i$s.
Then the sample correlation $r_{xy} = \frac{S_{xy}}{S_xS_y}.$ Multiplying the numerator and denominator
both by $n-1$, we see that $r_{xy}=U$ of your question.
In R:
cov(x,y); sd(x); sd(y)
[1] 0.78
[1] 10.70825
[1] 0.07924645
u = cov(x,y)/(sd(x)*sd(y)); u
[1] 0.9191707
cor(x,y)
[1] 0.9191707

plot(x,y, pch=20)


In the second part, for increasingly large samples, the sample statistics
$S_x, S_y, S_{xy}$ converge to their respective population
counterparts $\sigma_x, \sigma_y, \sigma_{xy}.$
Then (provided $\sigma_x > 0, \sigma_y > 0)$ it follows that $r_{xy}$ converges to $\rho_{xy}.$ (Slutsky's Theorem.)
As an example let $X\sim\mathsf{Norm}(\mu=50,\sigma=5)$ and, independently,
$Z \sim\mathsf{Norm}(0,1).$
Also, let $Y = 50+2X+Z.$
Then it is easy to show that $Var(X) = 25,\,$
$Var(Y) = 101,\,$ $Cov(X,Y) = 50,\,$ and
$\rho_{xy}=Cor(X,Y)$ $= \frac{50}{2\sqrt{101}}$ $= 0.99504.$
Simulating samples of ten million, we see that for
large $n$ we have $r_{xy} \approx 0.99504.$
set.seed(112)
n=10^7
x = rnorm(n, 50, 5)
y = 50 + 2*x + rnorm(n, 0, 1)
var(x); var(y); cov(x,y); cor(x,y)  
[1] 24.99581    # aprx 25
[1] 100.9826    # aprx 101
[1] 49.99143    # aprx 50
[1] 0.9950357   # aprx 0.99594
cov(x,y)/(sd(x)*sd(y))
[1] 0.9950357

The plot below shows the first 5000 simulated
$(X_i,Y_i).$

