Abstract
In an earlier paper, we proved that a plot of log n independent Brownian motions in dimension d=1, for times in [0,n], is nearly certain to give the appearance of a shaded region with square root boundaries, when subjected to the rescaling of the functional iterated logarithm law. Here we prove that for every finite dimension d the same conclusion holds if just one point is plotted from each of log n Brownian paths having variance parameter equal to 1, provided these points are selected uniformly in the time interval [0,n] and independently of the paths.
Keywords
- Brownian Motion
- Variance Parameter
- Shade Region
- Finite Dimension
- Iterate Logarithm
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References
LePage, R. and Schreiber, B. M. An iterated logarithm law for families of Brownian paths. To appear, Z. W. Verw. Geb. (1984).
Richardson, L. F. Some measurements of atmospheric turbulance. Phil. Trans. Roy. Soc. Lond, A. Vol. 221 (1921), p. 1.
Taylor, G. I. Diffusion by continuous movements. Proc. London Math. Soc. 20 (1921), 196–211.
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© 1985 Springer-Verlag
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LePage, R., Schreiber, B.M. (1985). A square root law for diffusing particles. In: Beck, A., Dudley, R., Hahn, M., Kuelbs, J., Marcus, M. (eds) Probability in Banach Spaces V. Lecture Notes in Mathematics, vol 1153. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074958
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DOI: https://doi.org/10.1007/BFb0074958
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-15704-5
Online ISBN: 978-3-540-39645-1
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