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Title:Some Almost Sure Convergence Results
Author(s):Fisher, Evan David
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Mathematics
Abstract:The first chapter of the thesis consists of an almost sure invariance principle for random variables in the Domain of Attraction of a Stable Law. Let {Y,Y(,1),Y(,2),...} be a sequence of i.i.d. symmetric random variables with Y in the domain of attraction of X where X is symmetric and stable of index (alpha). Suppose {a(,n)}, 0 ) 0 as i (--->) (INFIN). This extends an analogous result of Stout's where the more restrictive assumption is made that Y is in the domain of normal attraction of X.
The second chapter of the thesis contains an upper class law of the iterated logarithm for supermartingales, with hypotheses analogous to the Kolmogorov Law of the Iterated Logarithm. Let {U(,n), (,n), n (GREATERTHEQ) 1} be a supermartingale with X(,n) = U(,n) - U(,n-1). Define
(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
and (theta)(,n) = (2 log(,2)s(,n)('2))(' 1/2) a.s. for n (GREATERTHEQ) 1 with s(,n) (--->) (INFIN) a.s. Suppose X(,i) (LESSTHEQ) K(,i)s(,i)/(theta)(,i) a.s. for each i (GREATERTHEQ) 1 where
(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
K(,i) is (,i-1)-measurable and K (GREATERTHEQ) 1/2. A function (epsilon)((.)) is given so that
(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
This result extends one of Stout's where he assumes 0 ) 0 as K (--->) 0 thus containing the Kolmogorov Law of the Iterated Logarithm as a special case.
In the third chapter of the thesis, two theorems are proved concerning normed weighted averages of a sequence of i.i.d. random variables. Let {Y,Y(,i), i (GREATERTHEQ) 1} be a sequence of i.i.d. random variables and let a(,j) > 0 with
(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
Issue Date:1981
Type:Text
Language:English
Description:66 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981.
URI:http://hdl.handle.net/2142/68186
Other Identifier(s):(UMI)AAI8203458
Date Available in IDEALS:2014-12-14
Date Deposited:1981


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