Invariance principles for adaptive self-normalized partial sums processes

Alfredas Rackauskas and Charles Suquet

Stochastic Processes and their Applications, 2001, vol. 95, issue 1, 63-81

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.

Keywords: Functional; central; limit; theorem; Domain; of; attraction; Holder; space; Randomization (search for similar items in EconPapers)
Date: 2001
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Citations: View citations in EconPapers (4)

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