A characterization of the innovations of first order autoregressive models

© 2014, Springer-Verlag Berlin Heidelberg. Suppose that $$Y_t$$Yt follows a simple AR(1) model, that is, it can be expressed as $$Y_t= \alpha Y_{t-1} + W_t$$Yt=αYt-1+Wt, where $$W_t$$Wt is a white noise with mean equal to $$\mu $$μ and variance $$\sigma ^2$$σ2. There are many examples in practice where these assumptions hold very well. Consider $$X_t = e^{Y_t}$$Xt=eYt. We shall show that the autocorrelation function of $$X_t$$Xt characterizes the distribution of $$W_t$$Wt.