19961201
Weighted approximations of partial sum processes in D[0, ∞). I
Publication
Publication
Studia Scientiarum Mathematicarum Hungarica , Volume 31  Issue 13 p. 323 353
Let X1, X2, . . . be independent, identically distributed random variables with S(k) = X1+. . .+ Xk. We prove best possible weighted approximations of n1/2S(nt), 0 t ≦1, by a standard Wiener process {W(i), 0 ≦ t < ∞} assuming the existence of two moments only for X1 and, consequently, obtain the weighted version of Donsker's theorem in D[0, 1] for the optimal class of weight functions. Considering functions h on [1, ∞) such that lim sup W(t)/h(t) < ∞ a.s. enables us to prove weighted approximations, and hence t→∞ also weak convergence, of weighted partial sum processes in D[1, ∞). In this case the admissible class of weight functions will be seen to be bigger than that for asymptotics on [0,1]. We show also that the class of weight functions for weighted sup and Lpfunctionals to converge in distribution is larger than that for weak convergence. Our proofs are based on Theorem of Major [17].
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Studia Scientiarum Mathematicarum Hungarica  
Organisation  School of Mathematics and Statistics 
Szyszkowicz, B. (1996). Weighted approximations of partial sum processes in D[0, ∞). I. Studia Scientiarum Mathematicarum Hungarica, 31(13), 323–353.
