Invariance Principles for Random Processes Generated by Extrema and Partial Sums of Random Variables
Dabrowski, Andre Robert
This item is only available for download by members of the University of Illinois community. Students, faculty, and staff at the U of I may log in with your NetID and password to view the item. If you are trying to access an Illinois-restricted dissertation or thesis, you can request a copy through your library's Inter-Library Loan office or purchase a copy directly from ProQuest.
Permalink
https://hdl.handle.net/2142/71206
Description
Title
Invariance Principles for Random Processes Generated by Extrema and Partial Sums of Random Variables
Author(s)
Dabrowski, Andre Robert
Issue Date
1982
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
Assume that F is a distribution function such that for some non-degenerate distribution function G, and sequences of constants a(,n) (a(,n) positive) and b(,n), we have that, as n increases, F('n)(a(,n)x + b(,n)) converges to G(x) for every real value x. Under these conditions we prove a strong invariance principle for extremal processes analogous to the well-known invariance principle of Strassen for partial sums. In our case, the strong invariance principle yields several weak convergence results for extremal processes. This differs from the case of partial sums in that Strassen's result does not yield the central limit theorem. By a similar procedure, we also obtain a strong invariance principle for point processes in the plane.
Using a different method, we prove an invariance principle in probability for extremal processes arising from stationary sequences satisfying certain weak dependence conditions. Of independent interest is a generalization to dependent sequences of a technique of Major used to obtain an universal approximating sequence from a collection of sequences, each of which approximates the desired sequence only to within a fixed positive tolerance.
We improve a strong invariance principle for partial sums of a stationary (phi)-mixing sequence of random variables with finite (2 + (delta)) moment (0 < (delta) (LESSTHEQ) 2) due to Berkes and Philipp. The rate of decay, (phi)(n), can be as slow as log n to the power -(1 + (epsilon)) (1 + 2/(delta)), where (epsilon) is some positive value. The error term obtained is sufficient to yield the central limit theorem and upper and lower class refinements to the law of the iterated logarithm.
The generalization of a technique of Major mentioned earlier is used to improve an invariance principle in probability due to Philipp for partial sums of (phi)-mixing sequences of Banach space valued random variables in the domain of attraction to a Gaussian law.
Use this login method if you
don't
have an
@illinois.edu
email address.
(Oops, I do have one)
IDEALS migrated to a new platform on June 23, 2022. If you created
your account prior to this date, you will have to reset your password
using the forgot-password link below.
