Persistence exponents via perturbation theory: Gaussian 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 standard normally distributed random variables, we study the persistence probabilities $\mathbb P(X_0\ge 0,\ldots , X_N\ge 0)$ for $N\to \infty $. We exploit the fact 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.

Published in Open Access (under CC-BY license).