# When is the Correlation Coefficient Ergodic

Given Wide Sense Stationary (WSS) processes X and Y that are ergodic to the mean and autocovariance. Under what conditions is the correlation coefficient ergodic to the mean? ie: $$lim_{T->\infty} \frac{1}{T} \int_0^T X(\tau)Y(\tau) d\tau$$ converges to E[X(t)Y(t)]=constant

• What is the signal processing context in this question? – MBaz Apr 21 at 1:11
• @MBaz Oftentimes one estimates the correlation coefficient by averaging the correlation over time. For this to be valid it must be ergodic to the mean. Knowing this allows one to know when this is fine to do. – roobee Apr 21 at 1:13