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

TL;DR

This paper investigates the decay rate of persistence probabilities in MA(1) processes using functional analysis and perturbation theory, providing a power series expansion for the persistence exponent and extending results to the Slepian process.

Contribution

It introduces a novel perturbation approach to compute the persistence exponent for MA(1) processes and applies it to the Slepian process, connecting functional analysis with stochastic persistence.

Findings

Persistence exponent expressed as a power series in 

Method applicable to Slepian process via transformation

Eigenvalue approach for decay rate analysis

Abstract

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 $N\to\infty$. 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.