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Proceedings of the American Mathematical Society

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Multiple images and local times of measurable functions

Author: Simeon M. Berman
Journal: Proc. Amer. Math. Soc. 97 (1986), 328-330
MSC: Primary 60J55; Secondary 60G17
MathSciNet review: 835892
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Abstract: Let $ x(t)$, $ 0 \leq t \leq 1$, be a real-valued measurable function having a local time $ {\alpha _{[0,t]}}(x)$ which is continuous in $ t$, for almost all $ x$. Then, for every integer $ m \geq 2$, and every nonempty open subinterval $ J \subset [0,1]$, there exist $ m$ disjoint subintervals $ {I_1}, \ldots ,{I_m} \subset J$ such that the intersection of the images of $ {I_1}, \ldots ,{I_m}$ under the mapping $ t \to x(t)$ has positive Lebesgue measure. The result applies to a large class of sample functions of stochastic processes, and also to multidimensional $ t$ and $ x( \cdot )$.

References [Enhancements On Off] (What's this?)

  • [1] S. M. Berman, Nonincrease almost everywhere of certain measurable functions with applications to stochastic processes, Proc. Amer. Math. Soc. 88 (1983), 141-144. MR 691295 (84d:60065)
  • [2] -, Multiple images of stochastic processes, Math. Proc. Cambridge Philos. Soc. 94 (1983), 183-188. MR 704810 (84k:60058)
  • [3] D. Geman and J. Horowitz, Occupation densities, Ann. Probab. 8 (1980), 1-60. MR 556414 (81b:60076)
  • [4] N. R. Shieh, Multiple points of a random field, Proc. Amer. Math. Soc. 92 (1984), 279-282. MR 754721 (86c:60062)

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Keywords: Multiple image, local time
Article copyright: © Copyright 1986 American Mathematical Society

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