On weighted approximations in D[0, 1] with applications to self-normalized partial sum processes

• Published in: Acta mathematica Hungarica Vol. 121; no. 4; pp. 307 - 332
• Main Authors: Csörgő, M., Szyszkowicz, B., Wang, Q.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2008
• Subjects: Mathematics; Mathematics and Statistics; 60F25; secondary 62E20; functional central limit theorems; weighted approximations in probability; domain of attraction of the normal law; approximations; primary 60G50; 60F17; self-normalized sums
• ISSN: 0236-5294; 1588-2632
• DOI: 10.1007/s10474-008-7216-5

Let X , X 1 , X 2 ,… be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in D [0, 1] for the partial sum processes { S [ nt ] , 0 ≦ t ≦ 1} where S n = Σ j =1 n X j , under the assumption that X belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes { S [ nt ] = V n , 0 ≦ t ≦ 1}, where V n 2 = Σ j =1 n X j 2 . L p approximations of self-normalized partial sum processes are also discussed.