MA(1) processes with Laplace innovations conditioned to stay positive

Abstract

We study a moving-average process with (not necessarily symmetric) Laplace innovations under the constraint of positivity. In the three nondegenerate parameter regimes $0<θ<1$, $θ>1$, and $θ<0$, we prove convergence of the conditioned finite-dimensional distributions and identify the limit as a Doob $h$-transform. The regimes lead to qualitatively different limiting dynamics: the invariant law is supported on the positive half-line for $0<θ<1$, the conditioned chain is confined to the negative half-line for $θ>1$, and the dynamics are genuinely two-sided and governed by $q$-trigonometric functions for $θ<0$. In each case, the persistence exponent, sharp persistence asymptotics, the defining eigenfunction, and the unique invariant distribution are obtained explicitly.