Persistence exponents via perturbation theory: MA(1)-processes
For the moving average process $X_n=ρξ_{n-1}+ξ_n$, $n\in\mathbb{N}$, where $ρ\in\mathbb{R}$ and $(ξ_i)_{i\ge -1}$ 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 $λ_ρ$ 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 $λ_ρ$ can be expressed as a power series in $ρ$. 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.