Multiple points of a random field
HTML articles powered by AMS MathViewer
- by Narn Rueih Shieh PDF
- Proc. Amer. Math. Soc. 92 (1984), 279-282 Request permission
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.References
- Simeon M. Berman, Local nondeterminism and local times of general stochastic processes, Ann. Inst. H. Poincaré Sect. B (N.S.) 19 (1983), no. 2, 189–207. MR 700709 —, Multiple images of stochastic processes, preprint.
- 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
- Jack Cuzick, Multiple points of a Gaussian vector field, Z. Wahrsch. Verw. Gebiete 61 (1982), no. 4, 431–436. MR 682570, DOI 10.1007/BF00531614
- Donald Geman and Joseph Horowitz, Occupation densities, Ann. Probab. 8 (1980), no. 1, 1–67. MR 556414
- André Goldman, Points multiples des trajectoires de processus gaussiens, Z. Wahrsch. Verw. Gebiete 57 (1981), no. 4, 481–494 (French). MR 631372, DOI 10.1007/BF01025870
- W. J. Hendricks, Multiple points for transient symmetric Lévy processes in $\textbf {R}^{d}$, Z. Wahrsch. Verw. Gebiete 49 (1979), no. 1, 13–21. MR 539660, DOI 10.1007/BF00534336
- Norio Kôno, Double points of a Gaussian sample path, Z. Wahrsch. Verw. Gebiete 45 (1978), no. 2, 175–180. MR 510534, DOI 10.1007/BF00715191
- Loren D. Pitt, Local times for Gaussian vector fields, Indiana Univ. Math. J. 27 (1978), no. 2, 309–330. MR 471055, DOI 10.1512/iumj.1978.27.27024
- Jay Rosen, Self-intersections of random fields, Ann. Probab. 12 (1984), no. 1, 108–119. MR 723732 N. R. Shieh, Joint continuity of local times for random fields, preprint.
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 92 (1984), 279-282
- MSC: Primary 60G17; Secondary 60G15, 60G60
- DOI: https://doi.org/10.1090/S0002-9939-1984-0754721-2
- MathSciNet review: 754721