Covariance of sums of pairs of correlated variables

Take two vectors of normally-distributed random variables

$$\mathbf{x} = (x_1, x_2, \ldots x_n)$$

$$\mathbf{y} = (y_1, y_2, \ldots y_n)$$

where the covariance of each pair $$(x_i, y_i)$$ is known,

$$\mathbf{C}_{x_{i}y_{i}} = \pmatrix{\sigma^{2}_{x_i} & \sigma^{2}_{x_{i}y_{i}}\\ \sigma^{2}_{x_{i}y_{i}}& \sigma^{2}_{y_i}}$$ for $$i = 1, 2, \ldots n$$,

and all other covariances are zero. The sum of each vector is

$$S_{x} = \Sigma_{i}x_i$$

$$S_{y} = \Sigma_{i}y_i$$

What is the correlation matrix $$(C_{S_{x}S_{y}})$$ of the sums $$S_{x}$$ and $$S_{y}$$ in terms of the $$\sigma^{2}_{x_i}$$, $$\sigma^{2}_{y_i}$$ and $$\sigma^{2}_{x_{i}y_{i}}$$?

• I presume at the end that one of the "$S_{x_i}$" is intended to be "$S_{y_i}.$" The notation is confusing, though: what does the "$i$" mean in this terminology? Moreover, since the sums appear to be numbers, in what sense would they have a "correlation matrix"?
– whuber
Jan 20 '19 at 22:07
• I tried to make it clearer. Jan 21 '19 at 15:23

I worked it out. The covariance between the two sums is the sum of the covariances of the pairs.

$$C_{S_{x}S_{y}} \equiv \mathrm{Cov}(S_x,S_y) = \mathrm{Cov}(x_1 + x_2 + \ldots + x_n, y_1 + y_2 + \ldots + y_n)$$

From bilinearity of covariance:

$$=\mathrm{Cov}(x_1, y_1) + \mathrm{Cov}(x_2, y_1) + \ldots \mathrm{Cov}(x_n, y_1)$$

$$+\mathrm{Cov}(x_1, y_2) + \mathrm{Cov}(x_2, y_2) + \ldots \mathrm{Cov}(x_n, y_2)$$

$$\ldots$$

$$+\mathrm{Cov}(x_1, y_n) + \mathrm{Cov}(x_2, y_n) + \ldots \mathrm{Cov}(x_n, y_n)$$

Because only each pair is correlated:

$$=\mathrm{Cov}(x_1, y_1) + \mathrm{Cov}(x_2, y_2) + \ldots \mathrm{Cov}(x_n, y_n)$$