A first-order autoregressive model for the joint distribution of a time dependent random variable and its derivative

Abstract

A first-order autoregressive (AR(1)) model is analysed to investigate the relationship between a time dependent random variable Γt and its derivative . This is motivated by the problem of the turbulent dispersion of a gas cloud or plume in the atmosphere, where the square of the derivative, , is also of interest. It is shown that in general Γt and Dt are correlated, with their correlation proportional to the skewness of Γt. The probability distributions of the innovations and of Yt and Dt, and the joint distributions of Γt and Yt, and of Γt and Dt, are derived for a number of assumed distributions for Γt. The latter include the normal, exponential, gamma (of integer order), uniform, and sum of uniform distributions. The lack of time reversibility of the AR(1) model is apparent in the asymmetric distributions of the derivative when Γt has the exponential or gamma distribution. Comparisons are made with some experimental observations. Copyright © 2011 John Wiley & Sons, Ltd.