# Auto/cross correlation for data set with multiple realizations

Context

For an assignment, I need to find the correlation matrix of a dataset with 100 random variables and 1000 realizations for each. (It's a stock market example).

Problem

I don't understand how one can compute the auto/cross-correlation for a given dataset of discrete random variables with multiple realizations.

Example

If $$x(n)$$ is a random variable with three realizations. E.g. (some example I came up with):

$$x(n) = \begin{Bmatrix} 2, & 1, & 3 \end{Bmatrix}$$

Then how do you compute its autocorrelation $$r_{xx}$$?

I know that $$r_{xx} = E[x(n)x^*(n-k)]$$, which will result in a convolution function. I think it should be something like:

$$\sum x(n)x(n-k) P(x=a_k,y=b_k)$$ $$\rightarrow$$ $$\frac{1}{N^2}\sum x(n)x(n-k)$$

Because the probability of it having some set of values is $$1/N^2 = 1/3^2$$, I think.

However, I think this is completely wrong because this gives an array of values and I also tried computing a correlation matrix with $$c_{xx} = r_{xx} - m_x^2$$, and that gives different results.

I just really want to understand how it works, but I haven't been able to. I would really appreciate some help on this matter.

Say $$x(n) = \begin{Bmatrix} (n=0) & (n=1) & (n=2) & (n=3) \\ 2 & 1 & 3 & 4\\ 1 & 2 & 1 & 3 \\ 1 & 3 & 2 & 3 \end{Bmatrix}$$