Some Limit Theorems (Empirical Processes)

Lacey, Michael Thoreau

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Description

Title

Some Limit Theorems (Empirical Processes)

Author(s)

Lacey, Michael Thoreau

Issue Date

1987

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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 real-valued 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 complex-valued functions on ($-1,1$).

Graduation Semester

1987

Type of Resource

text

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http://hdl.handle.net/2142/71257

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