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Uniform quadratic variation for Gaussian processes

Author

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  • Adler, Robert J.
  • Pyke, Ron

Abstract

We study the uniform convergence of the quadratic variation of Gaussian processes, taken over large families of curves in the parameter space. A simple application of our main result shows that the quadratic variation of the Brownian sheet along all rays issuing from a point in [0, 1]2 converges uniformly (with probability one) as long as the meshes of the partitions defining the quadratic variation do not decrease too slowly. Another application shows that previous quadratic variation results for Gaussian processes on [0, 1] actually hold uniformly over large classes of partitioning sets.

Suggested Citation

  • Adler, Robert J. & Pyke, Ron, 1993. "Uniform quadratic variation for Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 48(2), pages 191-209, November.
    • Handle: RePEc:eee:spapps:v:48:y:1993:i:2:p:191-209
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    Citations

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    Cited by:

    1. Guyon, Xavier & Perrin, Olivier, 2000. "Identification of space deformation using linear and superficial quadratic variations," Statistics & Probability Letters, Elsevier, vol. 47(3), pages 307-316, April.
    2. Istas, Jacques, 2007. "Quadratic variations of spherical fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 117(4), pages 476-486, April.
    3. Benassi, Albert & Cohen, Serge & Istas, Jacques & Jaffard, Stéphane, 1998. "Identification of filtered white noises," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 31-49, June.
    4. Perrin, Olivier, 1999. "Quadratic variation for Gaussian processes and application to time deformation," Stochastic Processes and their Applications, Elsevier, vol. 82(2), pages 293-305, August.

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