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

For the moving average process Xₙ=ρ ξₙ₋₁+ξₙ, n, where ρ and (ξᵢ)i≥ -1 is an i.i.d. sequence of normally distributed random variables, we study the persistence probabilities P(X₀≥ 0,, XN≥ 0), for N→∞. 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.