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The Reduction Principle for the Empirical Process of a Long Memory Linear Process

  • Liudas Giraitis
  • Donatas Surgailis

Abstract

We discuss the uniform reduction principle for the empirical process of a long memory moving average process X t, t ∈ ℤ with long memory, which generalizes the corresponding reduction principle of Dehling and Taqqu [9]. The proof is based on the expansion of the bivariate probability density t (x 1, x 2) of X 0, X t :
$$ {{f}_{1}}({{x}_{1}},{{x}_{2}}) = f({{x}_{1}})f({{x}_{2}}) + {{r}_{t}}f'({{x}_{1}})f'(x2) + o({{r}_{t}}),\quad t \to \infty , $$
uniformly in x 1, x 2, where r t = (X 0, X t ) and f(x) is the marginal probability density. An easy consequence of the reduction principle is the functional CLT for the empirical process. An application of the last result to the change-point problem of the marginal c.d.f. is discussed.

Keywords

Central Limit Theorem Fractional Brownian Motion Linear Process Empirical Process Gaussian Case 
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Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Liudas Giraitis
    • 1
  • Donatas Surgailis
    • 2
  1. 1.Department of EconomicsLondon School of EconomicsLondonUK
  2. 2.Vilnius Institute of Mathematics and InformaticsVilniusLithuania

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