The mixed exponential solution to the first-order autoregressive model

This paper gives the distribution of the independent variable in a first-order autoregressive equation which is necessary for the modelled variable to have a mixed exponential marginal distribution. Restricting attention first to a mixture of two exponentials which is probabilistic, a solution is shown to exist over a restricted range of the first serial correlation; tabulations of the boundary values are given. The independent variable has a distribution which includes a discrete component at zero and a not necessarily probabilistic mixture of three exponentials. The marginal distribution is then considered with a negative mixture weight; this weight is shown to be lower bounded in value, but not to depend on the first serial correlation.