A note on the innovation distribution of a gamma distributed autoregressive process (Q2742768)

The author considers the first order autoregressive process \(X_n =p X_{n-1} +\varepsilon_n\), where \(0< p<1\) and \(\{ \varepsilon_n \}\) is an i.i.d. sequence of random variables. The following assertion is proved:NEWLINENEWLINENEWLINEConsider the random variable \(W\) defined as follows. \(W/ \Pi \sim \text{ga}(\Pi, 1 + \phi)\), \(\Pi/Z \sim \text{po}(\phi Z)\), \(Z \sim \text{ga}(a, 1)\), where \(\text{ga}(a, b)\) denotes a gamma distribution with mean \(a/b\) and \(p(\lambda)\) a Poisson distribution with mean \(\lambda.\) If \(X_{n-1} \sim \text{ga}(a, 1)\) and \(X_n = \rho X_{n-1} +W\) then marginally \(X_n \sim \text{ga}(a, 1).\) Here \(\rho =1/(1 +\phi)\). Note that if \(\Pi =0\), then \(P(W =0) =1.\)