Rates of convergence in a central limit theorem for stochastic processes defined by differential equations with a small parameter

• Author(s) / Creator(s)

  • Kouritzin, Michael
  • Heunis, A.J.

• Let μ be a positive finite Borel measure on the real line R. For t ≥ 0 let et · E1 and E2 denote, respectively, the linear spans in L2(R, μ) of {eisx, s > t} and {eisx, s < 0}. Let θ: R → C such that ∥θ∥ = 1, denote by αt(θ, μ) the angle between θ · et · E1 and E2. The problems considered here are that of describing those measures μ for which (1) αt(θ, μ) > 0, (2) View the MathML source as t → ∞ (such μ arise as the spectral measures of strongly mixing stationary Gaussian processes), and (3) give necessary and sufficient conditions for the rate of convergence of the generalized maximal correlation coefficient: ϱt(θ, μ) = cos αt(θ, μ). Using this coefficient we characterize the stationary continuous processes that are (a) completely regular and (b) strongly mixing Gaussian. We also give necessary and sufficient conditions for the rate of convergence of (a) the maximal correlation coefficient and (b) the mixing coefficient in the Gaussian case.

• Date created

  1992

• Subjects / Keywords

  • Averaging principle
  • Functional central limit theorem
  • Prohorov distance
  • Strong mixing non-stationary stochastic processes

• Type of Item

  Article (Published)

• DOI

  https://doi.org/10.7939/R36Q1SP6Q

• License

  © 1997 Journal of Multivariate Analysis. This version of this article is open access and can be downloaded and shared. The original author(s) and source must be cited. Non-commercial use only.