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Fluctuation theory of random walks by pairing paths

Fluctuation theory of random walks by pairing paths

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When Mar 27, 2019
from 12:45 pm to 02:00 pm
Speaker Robert Cordery PhD
Speaker Information Dr. Robert Cordery received his PhD in theoretical statistical physics from University of Toronto. His early research centered on renormalization group methods for statistical mechanics and field theory. After a Postdoc at Rutgers and a visiting position at Northeaster, he left academia for a 30 year career in the R&D group at Pitney Bowes. There he was named inventor on over 150 US patents. Bob is now Visiting Assistant Professor in the Physics Department at Fairfield University. Interest in the effect of light scattering on print quality led eventually to a new approach for proving and extending the almost forgotten results of fluctuation theory of random walks.
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Abstract:

Many statistical properties of discrete-time random walks on \(\mathbb{R}\) are surprisingly independent of the distribution of step-size. Distribution-free (DF) probabilities of symmetric walks include first passage time, time when the maximum is reached, ascending ladder epochs, and number of steps on the positive half-line. We present a powerful new heuristic that provides short, intuitive proofs of DF probabilities for certain random walk events. Our heuristic, based on considering pairs of walks, reproduces Fluctuation Theory results developed in the 1960's for symmetric walks that were proven using a variety of analytic, combinatoric and algebraic methods. Our DF results extend to certain events in alternating random walks that reverse direction each step and in persistent walks where the step direction probability depends on the previous step direction. The combinatoric form of the probabilities of these events resemble those of walks on lattices, not only in the limit of many steps, but even for short walks. In a related application of the heuristic, replacing walks on \(\mathbb{Z}\) by walks on \(\mathbb{R}\) with step-sizes perturbed around unity, we resolve ambiguity in the correct comparison of discrete and continuous walks.