Autocorrelation at Zero Lag for Strictly Stationary Stochastic Process is 1

Example of Strictly Stationary Stochastic Process

Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$.

Then the autocorrelation at zero lag is given by:

$\rho_0 = 1$

Proof

From Autocorrelation of Strictly Stationary Stochastic Process:

$\rho_k = \dfrac {\gamma_k} {\gamma_0}$

where $\gamma_k$ denotes the autocovariance of $S$.

For zero lag, $k = 0$.

Hence:

$\rho_0 = \dfrac {\gamma_0} {\gamma_0} = 1$

$\blacksquare$

Sources

• 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):

Part $\text {I}$: Stochastic Models and their Forecasting:

$2$: Autocorrelation Function and Spectrum of Stationary Processes:

$2.1$ Autocorrelation Properties of Stationary Models:

$2.1.2$ Stationary Stochastic Processes: Autocovariance and autocorrelation coefficients