On some distributional properties of a first-order nonnegative bilinear time series model (Q2774443)

The authors consider the nonnegative bilinear model \(X_t=Z_t+\lambda Z_t X_{t-1}\), where \(X_0\geq 0\), \(\{Z_t\}\) is a sequence of i.i.d. nonnegative random variables with uniform distribution in the interval \((0,1)\), and \(\lambda>0\) is a constant. It is proved that \(\{X_t\}\) is strictly stationary if \(\lambda \in (0,\text{e})\) and \(X_0\) has the same distribution as \(\sum_{m=1}^{\infty} Z_m \prod_{j=1}^{m-1} (\lambda Z_j)\). When \(\lambda\in (0,1)\), then \(P(0\leq X_0\leq (1-\lambda)^{-1})=1\); when \(\lambda=1\), then \(P(X_0>n)=O(n^{-n}\text{e}^n)\); when \(\lambda\in(1,\text{e})\), then \(P(X_0>x)\sim cx^{-\kappa}\) as \(x\to \infty\) for some positive constants \(c\) and \(\kappa\). Further, the moments and the covariance function of the stationary process \(\{X_t\}\) are calculated and a formula for the density of \(X_t\) is given.