University of Illinois Urbana-Champaign

Inequalities for Random Walk and Partially Observed Brownian Motion

Mcconnell, Terry Robert

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Description

Title
Inequalities for Random Walk and Partially Observed Brownian Motion
Author(s)
Mcconnell, Terry Robert
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
Abstract
  • This thesis is divided into two parts. The first part studies the control of the maximal function of N-dimensional 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
  • 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 N-dimensions (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.
Graduation Semester
1981
Type of Resource
text
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http://hdl.handle.net/2142/71197

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