Given two random sequences $X$ and $Y$ with the same length $n$, how can we infer the Pearson correlation coefficient (PCC) between them, e.g., the expected correlation coefficient?
When the two sequences are deterministic, the PCC would be $$ r(X, Y) = \frac{\sum_i (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum_i (X_i - \bar{X})^2 \sum_i (Y_i - \bar{Y})^2}}. $$ The results on special cases of $X$ and $Y$ would also be appreciated.