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Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

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  • Voit, Michael

Abstract

Forn[greater-or-equal, slanted]2, let ([mu]x[tau], n)[tau][greater-or-equal, slanted]0be the distributions of the Brownian motion on the unit sphereSn[subset of]n+1starting in some pointx[set membership, variant]Sn. This paper supplements results of Saloff-Coste concerning the rate of convergence of[mu]x[tau], nto the uniform distributionUnonSnfor[tau]-->[infinity] depending on the dimensionn. We show that,[formula]for[tau]n:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measures[mu]x[tau], nby measures which are known explicitly via Poisson kernels onSn, and which tend, after suitable projections and dilatations, to normal distributions on forn-->[infinity]. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups.

Suggested Citation

  • Voit, Michael, 1996. "Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions," Journal of Multivariate Analysis, Elsevier, vol. 59(2), pages 230-248, November.
  • Handle: RePEc:eee:jmvana:v:59:y:1996:i:2:p:230-248
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    References listed on IDEAS

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    1. Dielman, Terry E. & Rose, Elizabeth L., 1996. "A note on hypothesis testing in LAV multiple regression: A small sample comparison," Computational Statistics & Data Analysis, Elsevier, vol. 21(4), pages 463-470, April.
    2. Dielman, Terry E. & Rose, Elizabeth L., 1995. "A bootstrap approach to hypothesis testing in least absolute value regression," Computational Statistics & Data Analysis, Elsevier, vol. 20(2), pages 119-130, August.
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