My research is on "Coboundary" Theorems for collections of various types of random variables. These are extensions of results that have previously been published. In particular, K. Schmidt (1977) proved that if a strictly stationary sequence of real-valued random variables has the property that the family of distributions of its partial sums is tight, then the sequence is a coboundary, meaning that it is equal to the successive differences of another strictly stationary sequence. In 1995 my advisor, Professor Richard Bradley, removed the strict stationarity requirement from the construction of the new sequence, although preserving the result when the original sequence is strictly stationary. He further extended the result to C[0, 1]-valued random variables in 1997. There have also been various results for coboundary theorems with moment conditions, where the tightness assumption is replaced by the stronger condition that the partial sums be bounded in Lp for some p ≥ 1, with the additional conclusion that the terms of the new sequence have finite p-norms. This was done by Aaronson and Weiss in 2000 for p ∈ [1, ∞), and Bradley extended this to random fields, which are collections of random variables indexed by [special characters omitted]. In this context, the term "coboundary'' is reserved for another use, and hence we refer to this result under the name "cousin of coboundary".
The goal of my research has been to prove a coboundary-type result for C[0, 1]-valued random fields that includes moment conditions. These moment conditions involve Lp-boundedness of the sup-norms of the partial sums (which are partial rectangular sums in the case d > 1), as well as a moment condition on the modulus of continuity of the partial sums. In addition, the work includes similar results for different types of random variables, such as random elements of l2, random infinitely-differentiable functions, and 3 × 3 random matrices.
|Advisor:||Bradley, Richard C.|
|Commitee:||Goodman, Victor, Lindenstrauss, Ayelet, Torchinsky, Alberto|
|School Location:||United States -- Indiana|
|Source:||DAI-B 74/06(E), Dissertation Abstracts International|
|Keywords:||Coboundary theorems, Moment conditions, Random variables, Stationary sequences|
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