In this paper we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, characterize coincidence conditional probabilities terms (generalized) Doob transforms, which then leads to a stronger classification result for associated spectral radius and strong Liouville property. Further...

We calculate the Smith normal form of the adjacency matrix of each of the following graphs or their complements (or both): complete graph, cycle graph, square of the cycle, power graph of the cycle, distance matrix graph of cycle, Andrásfai graph, Doob graph, cocktail party graph, crown graph, prism graph, Möbius ladder. The proofs operate by finding the abelianisation of a cyclically presented...

* This work was supported by the United States Air Force under contract No. AF18(600)-685, monitored by the Office of Scientific Research. l Linnik, Yu. V., "Linear Statistics and the Normal Distribution Law," Doklady Akad. Nauk SSSR (N.S.), 83, 353-355, 1952; "On Some Identically Distributed Statistics," ibid., 89, 9-11, 1953. 2 Doob, J. L., Stochastic Processes (New York: John Wiley & Sons, 1...

•All have the property that E[θn+1 | θn] = θn. •Called the martingale property •Philosophically desirable because it means that information is preserved as we move down the hierarchy •Theorem (Doob): All non-negative martingale sequences have a limit with probability 1. θn+1 | θn ∼ DP(cθn), θn+1 | θn ∼ BP(cθn), θn+1 | θn ∼ GammaP(cθn), θn+1 | θn ∼ PYP(cθn) Pathological Properties of Deep Bayesi...

Wai Yan Pong ([email protected]) received his B.Sc. from the Chinese University of Hong Kong and his M.Sc. and Ph.D. from the University of Illinois at Chicago. He was a Doob Research Assistant Professor at the University of Illinois at Urbana-Champaign for three years. He then moved to California and is now teaching at California State University, Dominguez Hills. His research interests are in m...

In this paper, we consider R-valued integrable processes which are increasing in the convex order, i.e. R-valued peacocks in our terminology. After the presentation of some examples, we show that an R-valued process is a peacock if and only if it has the same one-dimensional marginals as an R-valued martingale. This extends former results, obtained notably by V. Strassen (1965), J.L. Doob (1968...