The autocorrelation function (ACF) is the normalized autocovariance function defined as $$ \rho (\tau)=\frac{\gamma(\tau)}{\sqrt{\gamma(0).\gamma(0)}}=\frac{\gamma(\tau)}{\gamma(0)}, $$where \(\tau = |s-t|\) is the lag time, or the amount of time by which the signal has been shifted.
The ACF measures the linear predictability of the series at time \( t \), say \(x_t \) using only the value \( \tau \). Using the Cauchy–Schwarz inequality which implies$$ {|\gamma(\tau)|}^2 \leq \gamma(0)^2 $$ it can be shown easily that$$ -1< \rho(\tau)<1. $$