Persistence exponents via perturbation theory: MA(1)-processes

For the moving average process $X_n=\rho {\xi }_{n-1}+{\xi }_n$, $n\in \mathbb{N}$, where $\rho \in \mathbb{R}$ and $({\xi }_i{)}_{i\ge -1}$ is an i.i.d. sequence of normally distributed random variables, we study the persistence probabilities $\mathbb{P}(X_0\ge 0,\dots ,X_N\ge 0)$, for $N\to \mathrm{\infty }$. We exploit that the exponential decay rate ${\lambda }_{\rho }$ of that quantity, called the persistence exponent, is given by the leading eigenvalue of a concrete integral operator. This makes it possible to study the problem with purely functional analytic methods. In particular, using methods from perturbation theory, we show that the persistence exponent ${\lambda }_{\rho }$ can be expressed as a power series in $\rho$. Finally, we consider the persistence problem for the Slepian process, transform it into the moving average setup, and show that our perturbation results are applicable.