A characterization of the innovations of first order autoregressive models

Abstract

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