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Extrema of some Gaussian processes with large trends and density estimation inL ∞ norm
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  • Published: September 1990

Extrema of some Gaussian processes with large trends and density estimation inL ∞ norm

  • V. D. Konakov1 

Probability Theory and Related Fields volume 86, pages 277–304 (1990)Cite this article

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Summary

Let {μ n (t),t ∈ [O,T n ]} be the sequence of trends andX n − μ n (t) is a stationary Gaussian process having mean zero. This paper establishes the exact asymptotic behaviour for the probabilities\(\Pr \mathop {(\max |X_n (t)|)}\limits_{[O, T_n ]} \leqq u_n ),\mathop {\lim }\limits_{x \to \infty } T_n = + \infty \), in the situation when the {μ n (t)} are not asymptotically negligible and essentially contribute in the approximating distribution. The results are used to study the optimal rates of convergence for nonparametric density estimation when the closeness is measured inL ∞ norm.

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Authors and Affiliations

  1. Central Economics-Mathematical Institute, Academy of Sciences, ul. Krasikova 32, 117418, Moscow, USSR

    V. D. Konakov

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  1. V. D. Konakov
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Konakov, V.D. Extrema of some Gaussian processes with large trends and density estimation inL ∞ norm. Probab. Th. Rel. Fields 86, 277–304 (1990). https://doi.org/10.1007/BF01208254

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  • Received: 22 March 1988

  • Revised: 17 March 1989

  • Issue Date: September 1990

  • DOI: https://doi.org/10.1007/BF01208254

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Keywords

  • Stochastic Process
  • Asymptotic Behaviour
  • Probability Theory
  • Density Estimation
  • Mathematical Biology
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