TY  - JOUR
AU  - Komlós, János
AU  - Major, Péter
AU  - Tusnády, Gábor
TI  - An approximation of partial sums of independent RV'-s, and the sample DF. I
JF  - ZEITSCHRIFT FUR WAHRSCHEINLICHKEITSTHEORIE UND VERWANDTE GEBIETE
J2  - Z WAHRSC VERW GEBIETE
VL  - 32
PY  - 1975
IS  - 1-2
SP  - 111
EP  - 131
PG  - 21
SN  - 0044-3719
DO  - 10.1007/BF00533093
UR  - https://m2.mtmt.hu/api/publication/100861
ID  - 100861
AB  - Let Sn=X1+X2+⋯+Xnbe the sum of i.i.d.r.v.-s, EX1=0, EX12=1, and let Tn= Y1+Y2+⋯+Ynbe the sum of independent standard normal variables. Strassen proved in [14] that if X1 has a finite fourth moment, then there are appropriate versions of Snand Tn(which, of course, are far from being independent) such that |Sn -Tn|=O(n1/4(log n)1/1(log log n)1/4) with probability one. A theorem of Bártfai [1] indicates that even if X1 has a finite moment generating function, the best possible bound for any version of Sn, Tnis O(log n). In this paper we introduce a new construction for the pair Sn, Tn, and prove that if X1 has a finite moment generating function, and satisfies condition i) or ii) of Theorem 1, then |Sn -Tn|=O(log n) with probability one for the constructed Sn, Tn. Our method will be applicable for the approximation of sample DF., too. © 1975 Springer-Verlag.
LA  - English
DB  - MTMT
ER  -