An Asymptotic Representation of the Sample Distribution Function

Abstract

distribution on [0, l]. Let Fn(x)~the proportion of the Xj^x. We will prove THEOREM. There is a random function {Gn(x); Orgarrgl}, with the same distribution as {Fn(x) ; 0 ^x g 1} for each n, and there is a Brownian motion W, such that for the Brownian B(x) =n ~ 1,2 W(nx) (1) ° sup | n l i*[Gn(x)-x\\- [B(x)- xB(l)] \\ M l almost surely as n—*<*>.- Ofri-^Oog w) 1/2 flog log n) l <*] This theorem is of use in the investigation of the asymptotic behavior of functionals of {Fn(x); 0^#^l}, especially functionals dependent on ». 2. We construct Gn(x) as follows; let Yu F2, • • • be independent exponential variables with mean 1. Let S(k) = Fi + • • • +F*t k = 1, 2, • • • and let 5(0) =0. Set Gn(x) « k/n HS(k)/S(n + 1) ^ x < S(k + 1)/S(n + 1). This {Gn(x) ; 0 ^x ^ 1} has the same distribution as {Fn(x) ; 0 £x S1} for each n. We now record a series of lemmas. LEMMA 1. There is a Brownian motion W such that (2) sup | *- S(k)- W(k) | = 0[n l »Qog w) 1/2 (loglogn) 1 '*] almost surely as n— • <». PROOF. This result is deducible from Theorem 1.5 of Strassen [8]. LEMMA 2. Almost surely as n— • <»