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On the quadratic moment of self-normalized sums

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  • Jonsson, Fredrik

Abstract

Let an integer n>=2 and a vector of independent, identically distributed random variables X1,...,Xn be given with and define the self-normalized sum . With a formula for we prove that and that if and only if the summands are symmetrically distributed. We also construct examples where Zn converges to the standard normal distribution as n tends to infinity while tends to infinity (the distribution of the summands varies with n).

Suggested Citation

  • Jonsson, Fredrik, 2010. "On the quadratic moment of self-normalized sums," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1289-1296, September.
    • Handle: RePEc:eee:stapro:v:80:y:2010:i:17-18:p:1289-1296
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    Cited by:

    1. Heiny, Johannes & Mikosch, Thomas, 2018. "Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices," Stochastic Processes and their Applications, Elsevier, vol. 128(8), pages 2779-2815.

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