In probability theory and statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is closely related to the autocorrelation of the process in question. Auto-covariance of stochastic processes Definition With the usual notation for the expectation operator, if the stochastic process has the mean function , then the autocovariance is given by where and are two instances in time. Definition for weakly stationary process If is a weakly stationary (WSS) process, then the following are true: for all and for all and where is the lag time, or the amount of time by which the signal has been shifted. The autocovariance function of a WSS process is therefore given by: which is equivalent to . Normalization It is common practice in some disciplines (e.g. statistics and time series analysis) to normalize the autocovariance function to get a time-dependent Pearson correlation coefficient. However in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "autocorrelation" and "autocovariance" are used interchangeably. The definition of the normalized auto-correlation of a stochastic process is . If the function is well-defined, its value must lie in the range , with 1 indicating perfect correlation and −1 indicating perfect anti-correlation. For a WSS process, the definition is . where . Properties Symmetry property respectively for a WSS process: Linear filtering The autocovariance of a linearly filtered process is Calculating turbulent diffusivity Autocovariance can be used to calculate turbulent diffusivity. Turbulence in a flow can cause the fluctuation of velocity in space and time. Thus, we are able to identify turbulence through the statistics of those fluctuations. Reynolds decomposition is used to define the velocity fluctuations (assume we are now working with 1D problem and is the velocity along direction): where is the true velocity, and is the expected value of velocity. If we choose a correct , all of the stochastic components of the turbulent velocity will be included in . To determine , a set of velocity measurements that are assembled from points in space, moments in time or repeated experiments is required. If we assume the turbulent flux (, and c is the concentration term) can be caused by a random walk, we can use Fick's laws of diffusion to express the turbulent flux term: The velocity autocovariance is defined as or where is the lag