Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, 1 1 1 - 131 (1975) 9 by Springer-Verlag 1975
An Approximation of Partial Sums of Independent RV'-s, and the Sample DF. I J. K o m l d s , P. M a j o r and G. Tusnfidy
Summary. Let S, = X x + X z + . . . + X, be the sum of i.i.d.r.v.-s, E X 1 =0, E X f = 1, and let T,= Y~+ Y2+ " ' + Y. be the sum of independent standard normal variables. Strassen proved in [14] that if X 1 has a finite fourth moment, then there are appropriate versions of S. and T, (which, of course, are far from being independent) such that IS.-T.l=O(n"(logn) ~1(log logn) § with probability one. A theorem of B/trtfai [1] indicates that even if X~ has a finite moment generating function, the best possible bound for any version of S,, T. is O(logn). In this paper we introduce a new construction for the pair S,, T,, and prove that if X~ has a finite moment generating function, and satisfies condition i) or ii) of Theorem 1, then IS, - T,[ = 0 (log n) with probability one for the constructed S,, T.. Our method will be applicable for the approximation of sample DF., too,
1. I n t r o d u c t i o n
Let S, = X~ + X 2 + . . - + X, be the sum of i.i.d.r.v.-s, with distribution function F(x), such that EX~ = 0, E X 2 = 1, and let W(t) be a Wiener process. The S k o r o h o d ' s embedding scheme (cf. [13]) provides a sequence of i.i.d.r.v.-s zj such that P(S~g.2M)+P\
3>~-,
, (2.10)
where the comma indicates the union of the events. Here the terms on the righthand side can be estimated by Tie ~'u-a'x with appropriate cq, 17~,Yi (i = 1, 2, 3, 4).
P(A l > e . 2 M ) < 2 N-M~ M~ sup
(P(Sk>e'2M)+p(skY'A) N~P(j~M(CI'2-j~j2-~-C2)>
X 4_ , A )
xI=P _xl (e xp { t (,~ t (C15+Cz)- 4 ) } >1 )
j) = ~ p(etX~>j)=oo, j=l
hence P ( l i m s u p Xj _>-1)=1. \ j+~o logj On the other hand
P/lim [Yj] =0~=1, \j~o logj
]
hence P (lim ~ p I S , - T,[> 1 logn = 2 t ) = 1 '
1
that is (1.2) holds with C O= 2~-" Similarly (1.2) holds with C O-
1
2 t if the moment
generating function of X 1 does not exist at some t
