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Multiple points of a random field

Author: Narn Rueih Shieh
Journal: Proc. Amer. Math. Soc. 92 (1984), 279-282
MSC: Primary 60G17; Secondary 60G15, 60G60
MathSciNet review: 754721
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Abstract: We prove that a $ d$-dimensional random field $ X \equiv {\{ X(t)\} _{t \in R_ + ^N}}$ has uncountably many $ r$-multiple points a.s. when it satisfies Pitt's ($ ({A_r})$) condition [9]. Those $ t$'s for which $ X(t)$ hits the multiple point can be separated by any given positive number, and multiple points can occur while $ t$ is restricted to any given "time inteval". Two corollaries concerning Gaussian fields and fields with independent increments are also presented.

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  • [1] S. M. Berman, Local nondeterminism and local times of general stochastic processes, preprint. MR 700709 (85b:60041)
  • [2] -, Multiple images of stochastic processes, preprint.
  • [3] R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments, J. Math. Mech. 10 (1961), 493-516. MR 0123362 (23:A689)
  • [4] J. Cuzick, Multiple points of a Gaussian vector field, Z. Wahrsch. Verw. Gebiete 61 (1982), 431-436. MR 682570 (84g:60075)
  • [5] D. Geman and J. Horowitz, Occupation densities, Ann. Probab. 8 (1980), 1-67. MR 556414 (81b:60076)
  • [6] A. Goldman, Points multiples des trajectoires de processus Gaussians, Z. Wahrsch. Verw. Gebiete 57 (1981), 481-494. MR 631372 (83a:60062)
  • [7] W. J. Hendricks, Multiple points for transient symmetric Lévy processes in $ {R^d}$, Z. Wahrsch. Verw. Gebiete 49 (1979), 13-21. MR 539660 (83c:60104)
  • [8] N. Kono, Double points of a Gaussian sample path, Z. Wahrsch. Verw. Gebiete 45 (1978), 175180. MR 510534 (80g:60033)
  • [9] L. D. Pitt, Local times for Gaussian vector fields, Indiana Univ. Math. J. 27 (1978), 309-390. MR 0471055 (57:10796)
  • [10] J. Rosen, Self-intersections of random fields, preprint. MR 723732 (85i:60052)
  • [11] N. R. Shieh, Joint continuity of local times for random fields, preprint.

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Keywords: Multiple points, local times, random fields, Pitt's condition
Article copyright: © Copyright 1984 American Mathematical Society

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