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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 Urbana-Champaign 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 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 ($-1,1$). Issue Date: 1987 Type: Text Description: 123 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987. URI: http://hdl.handle.net/2142/71257 Other Identifier(s): (UMI)AAI8721684 Date Available in IDEALS: 2014-12-16 Date Deposited: 1987
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