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A normal comparison inequality and its applications
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  • Published: April 2002

A normal comparison inequality and its applications

  • Wenbo V Li1 &
  • Qi-Man Shao2 

Probability Theory and Related Fields volume 122, pages 494–508 (2002)Cite this article

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  • 52 Citations

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Abstract

 Let $\xi=(\xi_i, 1 \leq i \leq n)$ and $\eta= (\eta_i, 1 \leq i \leq n)$ be standard normal random variables with covariance matrices $R^1=(r_{ij}^1)$ and $R^0=(r_{ij}^0)$, respectively. Slepian's lemma says that if $r_{ij}^1 \geq r_{ij}^0$ for $1 \leq i, j \leq n$, the lower bound $\P(\xi_i \leq u \mb{for} 1 \leq i \leq n ) /\P(\eta_i \leq u \mb{for} 1 \leq i \leq n ) $ is at least $1$. In this paper an upper bound is given. The usefulness of the upper bound is justified with three concrete applications: (i) the new law of the iterated logarithm of Erdős and Révész, (ii) the probability that a random polynomial does not have a real zero and (iii) the random pursuit problem for fractional Brownian particles. In particular, a conjecture of Kesten (1992) on the random pursuit problem for Brownian particles is confirmed, which leads to estimates of principal eigenvalues.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA. e-mail: wli@math.udel.edu. Supported in part by NSF Grant DMS-9972012., , , , , , US

    Wenbo V Li

  2. Department of Mathematics, University of Oregon, Eugene, OR 97403, USA. e-mail: shao@math.uoregon.edu. Supported in part by NSF Grant DMS-9802451., , , , , , US

    Qi-Man Shao

Authors
  1. Wenbo V Li
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  2. Qi-Man Shao
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Received: 17 October 2000 / Revised version: 30 May 2001 / Published online: 22 February 2002

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Li, W., Shao, QM. A normal comparison inequality and its applications. Probab Theory Relat Fields 122, 494–508 (2002). https://doi.org/10.1007/s004400100176

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  • Issue Date: April 2002

  • DOI: https://doi.org/10.1007/s004400100176

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Keywords

  • Covariance
  • Covariance Matrice
  • Brownian Particle
  • Iterate Logarithm
  • Normal Random Variable
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