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Invariance principles for adaptive self-normalized partial sums processes

Author

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  • Rackauskas, Alfredas
  • Suquet, Charles

Abstract

Let [zeta]nse be the adaptive polygonal process of self-normalized partial sums Sk=[summation operator]1[less-than-or-equals, slant]i[less-than-or-equals, slant]kXi of i.i.d. random variables defined by linear interpolation between the points (Vk2/Vn2,Sk/Vn), k[less-than-or-equals, slant]n, where Vk2=[summation operator]i[less-than-or-equals, slant]k Xi2. We investigate the weak Hölder convergence of [zeta]nse to the Brownian motion W. We prove particularly that when X1 is symmetric, [zeta]nse converges to W in each Hölder space supporting W if and only if X1 belongs to the domain of attraction of the normal distribution. This contrasts strongly with Lamperti's FCLT where a moment of X1 of order p>2 is requested for some Hölder weak convergence of the classical partial sums process. We also present some partial extension to the nonsymmetric case.

Suggested Citation

  • Rackauskas, Alfredas & Suquet, Charles, 2001. "Invariance principles for adaptive self-normalized partial sums processes," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 63-81, September.
    • Handle: RePEc:eee:spapps:v:95:y:2001:i:1:p:63-81
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    Citations

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    Cited by:

    1. Alfredas Račkauskas & Charles Suquet, 2004. "Necessary and Sufficient Condition for the Functional Central Limit Theorem in Hölder Spaces," Journal of Theoretical Probability, Springer, vol. 17(1), pages 221-243, January.
    2. Raluca Balan & Kulik, 2005. "Self-Normalized Weak Invariance Principle for Mixing Sequences," RePAd Working Paper Series lrsp-TRS417, Département des sciences administratives, UQO.
    3. Csörgő, Miklós & Martsynyuk, Yuliya V., 2011. "Functional central limit theorems for self-normalized least squares processes in regression with possibly infinite variance data," Stochastic Processes and their Applications, Elsevier, vol. 121(12), pages 2925-2953.
    4. Kulik, Rafal, 2006. "Limit theorems for self-normalized linear processes," Statistics & Probability Letters, Elsevier, vol. 76(18), pages 1947-1953, December.

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