Sample Autocorrelation

Definition

Let

• $z_1,z_2,\dots ,z_n$ : Random sample
• $\bar{z}=\frac{1}{n}{\sum }^n{z}_i$ : Sample mean of $z_1,\dots ,z_n$

The sample autocorrelation at lag $k$ is defined as

$$r_k=\frac{{\sum }_{t=k+1}^n(z_t-\bar{z})(z_{t-k}-\bar{z})}{{\sum }_{t=1}^n(z_t-\bar{z}{)}^2}$$