The sample autocorrelation function replaces the mean and lagged covariance in by their empirical counterparts.
White noise has constant mean, constant variance, and zero autocovariance at every nonzero lag. Gaussian white noise additionally has jointly Gaussian coordinates and is therefore independent across time.
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A stationary process is a stochastic (random) process whose statistical properties are invariant with respect to time. In other words, the joint probability distribution of the random variables in the process does not change when shifted in time. This means that the characteristics such as the mean, variance, and autocovariance remain constant over time.