stationary if since it is obtained as the output of a stable filter whose input is white noise. (If then the variance of depends on time lag t, so that the variance of the series diverges to infinity as t goes to infinity, and is therefore not weak sense stationary.) Assuming , the mean is identical for all values of t by the very definition of weak sense stationarity. If the mean is denoted by , it follows fromthatand hence The variance is where is the standard deviation of . This can be shown by noting that and then by noticing that the quantity above is a stable fixed point of this relation. The autocovariance is given by It can be seen that the autocovariance function decays with a decay time (also called time constant) of . The spectral density function is the Fourier transform of the autocovariance function. In discrete terms this will be the discrete-time Fourier transform: This expression is periodic due to the discrete nature of the , which is manifested as the cosine term in the denominator. If we assume that the sampling time () is much smaller than the decay time (), then we can use a continuum approximation to : which yields a Lorentzian profile for the spectral density: where is the angular frequency associated with the decay time . An alternative expression for can be derived by first substituting for in the defining equation. Continuing this process N times yields For N approaching infinity, will approach zero and: It is seen that is white noise convolved with the kernel plus the constant mean. If the white noise is a Gaussian process then is also a Gaussian process. In other cases, the central limit theorem indicates that will be approximately normally distributed when is close to one. For , the process will be a geometric progression (exponential growth or decay). In this case, the solution can be found analytically: whereby is an unknown constant (initial condition). Explicit mean/difference form of AR(1) process The AR(1) model is the discrete time analogy of the continuous Ornstein-Uhlenbeck process. It is therefore sometimes useful to understand the properties of the AR(1) model cast in an equivalent form. In this form, the AR(1) model, with process parameter is given by: , where and is the model mean. By putting this in the form , and then expanding the series