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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 Urbana-Champaign 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 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(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 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. Issue Date: 1981 Type: Text Description: 64 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981. URI: http://hdl.handle.net/2142/71197 Other Identifier(s): (UMI)AAI8203525 Date Available in IDEALS: 2014-12-16 Date Deposited: 1981
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