A note on the stationarity of a threshold first-order bilinear process

Abstract

In the present note we study the threshold first-order bilinear model X(t)=aX(t−1)+(b 11 {X(t−1)<c}+b 21 {X(t−1)⩾c})X(t−1)e(t−1)+e(t), tϵN where { e( t), tϵN} is a sequence of i.i.d. absolutely continuous random variables, X(0) is a given random variable and a, b 1, b 2 and c are real numbers. Under suitable conditions on the coefficients and lower semicontinuity of the densities of the noise sequence, we provide sufficient conditions for the existence of a stationary solution process to the present model and of its finite moments of order p.