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SOME LAW OF THE ITERATED LOGARITHM TYPE RESULTS FOR THE EMPIRICAL PROCESS
Author(s) -
Shorack Galen R.
Publication year - 1980
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1980.tb01153.x
Subject(s) - law of the iterated logarithm , logarithm , combinatorics , iterated function , mathematics , log log plot , distribution (mathematics) , order (exchange) , normalization (sociology) , iterated logarithm , binary logarithm , discrete mathematics , mathematical analysis , finance , sociology , anthropology , economics
Summary We consider Z ± n = sup 0< t ≤ 1/2 2 U ± n (t)/(t(1‐ t)) 1/2 , where + and ‐denote the positive and negative parts respectively of the sample paths of the empirical process U n . U ± n and U n are seen to behave rather differently, which is tied to the asymmetry of the binomial distribution, or to the asymmetry of the distribution of small order statistics. Csáki (1975) showed that log Z ± n /log 2 n is the appropriate normalization for a law of the iterated logarithm (LIL) for Z ± n we show that Z ‐ n /(2 log 2 n ) 1/2 is the appropriate normalization for Z ‐ n . Csörgö & Révész (1975) posed the question: if we replace the sup over (0,1/2) above, by ‐the sup over [a n , 1‐a n ] where a n →0, how fast can a n →0 and still have |Z n |/(2 log 2 n ) 1/2 maintain a finite lim sup a.s.? This question is answered herein. The techniques developed are then used in Section 4 to give an interesting new proof of the upper class half of a result of Chung (1949) for |U n (t)|. The proofs draw heavily on James (1975); two basic inequalities of that paper are strengthened to their potential, and are felt to be of independent interest.