# Auto-covariance function of station time series

How to show that for any stationary time series its auto-covariance function is symmetric about the origin, that is $$\gamma_{k}=\gamma_{-k}$$ where, $$\gamma_k=cov(z_t,z_{t-k})$$

Hi: Subtract $$k$$ from $$z_t$$ and add $$k$$ to $$z_{t-k}$$. Then you have $$cov(z_{t-k,} z_{t})$$ which by definition is $$\gamma_{-k}$$. But, by stationarity, this has to be equal to $$cov(z_{t}, z_{t-k})= \gamma_{k}$$ because the covariance is only a function of the lag difference.