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Title:  Some Limit Theorems (Empirical Processes) 
Author(s):  Lacey, Michael Thoreau 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  We establish a central limit theorem and bounded and compact laws of the iterated logarithm for partial sum processes indexed by classes of functions. For the central limit theorem, we assume an envelope condition and a majorizing measure condition more general than the usual metric entropy with bracketing. For the laws of the iterated logarithm, we assume an envelope condition and a growth condition on the metric entropy under bracketing. Examples show that our results are sharp. As a corollary we obtain new results for weighted sums of independent identically distributed random variables. Let X be a realvalued random variable with distribution function F(x) and characteristic function c(t). Let c$\sb{\rm n}$(t) be the characteristic function of F$\sb{\rm n}$(x), the nth empirical distribution function. We give necessary and sufficient conditions, in terms of c(t), for ${\rm n\sp{1/2}(c\sb{n}(t)}$ $$ c(t)) to obey bounded and compact laws of the iterated logarithm in C($1,1$), the Banach space of continuous complexvalued functions on ($1,1$). 
Issue Date:  1987 
Type:  Text 
Description:  123 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1987. 
URI:  http://hdl.handle.net/2142/71257 
Other Identifier(s):  (UMI)AAI8721684 
Date Available in IDEALS:  20141216 
Date Deposited:  1987 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois