Some topics in random walks on graphs, harmonic analysis and rogozin type inequalities for locally compact groups
Date
2017
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Publisher
University of Delaware
Abstract
This thesis lies in the intersection of areas like probability, harmonic analysis, combinatorics, and information theory. ☐ Specifically, the first part (probability and information theory) will present significant generalizations to locally compact groups of an inequality of Rogozin for convolutions on the real line. As corollaries, we obtained asymptotically sharp constants for infinity-Renyi entropy power inequalities for random variables taking values in either Rd or the integers. ☐ The second part (harmonic analysis) of this thesis will focus on the norm of the Fourier operator from Lp space to Lq space for compact or discrete abelian groups G, and for any possible pairs p and q. We will also examine the implications for uncertainty principles on such groups expressed in terms of Renyi entropies. ☐ The third part (combinatorics) of this thesis will analyze the mixing rates of random walks with possible backtracking on a generalized regular graph, which generalized the corresponding results about the simple or non-backtracking random walks on regular graphs. Moreover, some typical examples and compared the mixing rates for different kinds of random walks are also calculated.
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Keywords
Combinatorics, Harmonic analysis, Locally compact abelian group, Mixing rate, Probability, Random walk