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

For the moving average process Xn=ρξn−1+ξn, n∈N, where ρ∈R and (ξi)i≥−1 is an i.i.d. sequence of standard normally distributed random variables, we study the persistence probabilities P(X0≥0,…,XN≥0) for N→∞. We exploit the fact that the exponential decay rate λρ 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 λρ 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.