Autocorrelation function

Remarks

Every autocorrelation function satisfies $\rho(0) = 1$, $\rho(-h) = \rho(h)$ and $\abs{\rho(h)} \le 1$. It is nonnegative definite: $\sum_{i,j} a_i a_j \rho(t_i - t_j) \ge 0$ for all real $a_i$ and integers $t_i$, since this sum is $\operatorname{Var}(\sum_i a_i X_{t_i}) / \gamma(0)$.

Example: let $(\varepsilon_t)$ be uncorrelated with mean $0$ and variance $\sigma^2 > 0$, and let $X_t = \varepsilon_t + \varepsilon_{t-1}$. Then $\gamma(0) = 2\sigma^2$, $\gamma(\pm 1) = \sigma^2$ and $\gamma(h) = 0$ for $\abs{h} \ge 2$, so $\rho(\pm 1) = 1/2$ and $\rho(h) = 0$ for $\abs{h} \ge 2$.

Non-example: for a random walk $S_t = \varepsilon_1 + \cdots + \varepsilon_t$, the variance $t \sigma^2$ depends on $t$, so the process is not weakly stationary and $\rho$ is not defined.

The divisor $n$ in $\hat\gamma$, rather than $n - \abs{h}$, makes the sample autocovariance nonnegative definite. In signal processing, autocorrelation often means the unnormalized $E[X_{t+h} X_t]$.