University of Illinois Urbana-Champaign

Some Almost Sure Convergence Results

Fisher, Evan David

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Description

Title
Some Almost Sure Convergence Results
Author(s)
Fisher, Evan David
Issue Date
1981
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
Language
eng
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)
  • 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
  • K(,i) is (,i-1)-measurable and K (GREATERTHEQ) 1/2. A function (epsilon)((.)) is given so that
  • 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
Graduation Semester
1981
Type of Resource
text
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http://hdl.handle.net/2142/68186

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