# Equivalence between two expressions for autocorrelation

Have that $$\text{Corr}(X_t,X_{t+h}) = \frac{\text{Cov}(X_t,X_{t+h})}{\sqrt{\text{Var}(X_t)\text{Var}(X_{t+h})}}$$ and $$\rho(h) = \frac{\gamma_X(h)}{\gamma_X(0)}.$$ Those are both ways to express the ACF, so they're equivalent, right? And that would mean the denominators are the same for both, so $$\sqrt{\text{Var}(X_t)\text{Var}(X_{t+h})} = \gamma_X(0).$$ Is this equality here true always, even if the process is not stationary? Also, I'm not completely understanding where this equality comes from.

• What could "$\gamma_X(h)$" possibly mean for a non-stationary process?
– whuber
Apr 17, 2023 at 21:01
• @whuber that would just be Variance of the non stationary process? Apr 18, 2023 at 0:49
• @eddie, but that quantity is undefined when the process is nonstationary (except for some special cases). Apr 18, 2023 at 7:38
• @RichardHardy so if it's stationary, both formulas are same and give the right answer? If it's not stationary, then variance and covariance are not constant over time, so they're not defined, except for some special cases where the covariance and variance can be defined by some function dependent on time? Apr 18, 2023 at 11:54
• The ones you have in your post do not depend on time, so we cannot say they can be defined by some function dependent on time. Either these constant properties exist or they do not. (The ones that depend on time are conditional ones, and these are not the same as the ones you have specified.) Apr 18, 2023 at 12:03