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Research papers

Convergence and convergence rate to fractional Brownian motion for weighted random sums

T. Konstantopoulos^a, A. Sakhanenko<sup>b

a</sup> Department of Mathematics, University of Patras
^b Ugra State University

Abstract: We consider infinite sums of weighted i.i.d. random variables, with finite variance and arbitrary distribution, and derive a necessary and sufficient conditions for the weak convergence (in function space with uniform topology) of normalized sums to fractional Brownian motion (FBM). We consider also convergence rates questions. Using the embedding suggested by the Komlós–Major–Tusnády strong approximations method, we derive (under certain conditions on the weights) estimates for the quality of the functional approximation to FBM.

UDC: 519.214

MSC: 60F17, 60F15; 60G18, 60G15

Received September 25, 2004, published October 12, 2004

Language: English

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