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Title:  Inequalities for Random Walk and Partially Observed Brownian Motion 
Author(s):  Mcconnell, Terry Robert 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  This thesis is divided into two parts. The first part studies the control of the maximal function of Ndimensional Brownian motion, B(,t), by the maximal function of partially observed Brownian motion. Let R denote a fixed open subset of (//R)('N), G an arbitrary open subset, and T the first exit time of the Brownian motion from G. Define the maximal function, B(,T)('*), by (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) and the partially observed maximal function, B(,T)('*)(,(FDIAG)R), by (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) Let x(,0) be a fixed point not belonging to the closure of R and p a positive number. Then there is a constant C(,1) so that the inequality (1) E('x(,0))(B(,T)('*))('p) (LESSTHEQ) C(,1)E('x(,0))(B(,T)('*)(,(FDIAG)R))('p) holds as G varies provided that there exists a function u, harmonic in R, and constants C(,2) > 0 and q > p such that (2) (VBAR)x(VBAR)('q) (LESSTHEQ) u(x) (LESSTHEQ) C(,2)(VBAR)x(VBAR)('q) + C(,2), x (ELEM) R. Conversely, if (1) holds then so does (2) with q replaced by p. This result has applications in complex analysis and probability. The second part considers the integrability of exit times of random walks in Ndimensions (N (GREATERTHEQ) 2). Let S(,n) = X(,1) + X(,2) + ... + X(,n) be the n('th) partial sum of independent, identically distributed random vectors, X(,1),X(,2),..., having mean zero, finite second moments, and covariance matrix equal to the identity. Let W be an open subset of (//R)('N) which is invariant under positive dilations, and (tau) the first exit time of S(,n) from W. If the boundary of W satisfies certain regularity conditions then the range of exponents p for which (tau)(' 1/2) has a finite p('th )moment is essentially the same as the corresponding range for standard Brownian motion. 
Issue Date:  1981 
Type:  Text 
Description:  64 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1981. 
URI:  http://hdl.handle.net/2142/71197 
Other Identifier(s):  (UMI)AAI8203525 
Date Available in IDEALS:  20141216 
Date Deposited:  1981 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois