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

Abstract

For the moving average process $X_n=\ddot{I}\hat{I}{¾}_{n-1}+\hat{I}{¾}_n$, $n\in \mathbb{N}$, where $\ddot{I}\in \mathbb{R}$ and $(\hat{I}{¾}_i{)}_{i\ge -1}$ 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 $N\to \infty$. We exploit that the exponential decay rate $\hat{I}{»}_{\ddot{I}}$ 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 $\hat{I}{»}_{\ddot{I}}$ can be expressed as a power series in $\ddot{I}$. 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.

27 pages