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Estimation of the transition density of a Markov chain

Mathieu Sart (2014)

Annales de l'I.H.P. Probabilités et statistiques

We present two data-driven procedures to estimate the transition density of an homogeneous Markov chain. The first yields a piecewise constant estimator on a suitable random partition. By using an Hellinger-type loss, we establish non-asymptotic risk bounds for our estimator when the square root of the transition density belongs to possibly inhomogeneous Besov spaces with possibly small regularity index. Some simulations are also provided. The second procedure is of theoretical interest and leads...

Existence of graphs with sub exponential transitions probability decay and applications

Clément Rau (2010)

Bulletin de la Société Mathématique de France

In this paper, we recall the existence of graphs with bounded valency such that the simple random walk has a return probability at time n at the origin of order exp ( - n α ) , for fixed α ∈ [ 0 , 1 [ and with Følner function exp ( n 2 α 1 - α ) . This result was proved by Erschler (see [4], [3]); we give a more detailed proof of this construction in the appendix. In the second part, we give an application of the existence of such graphs. We obtain bounds of the correct order for some functional of the local time of a simple random walk on...

Fuzzy Markov chains: uncertain probabilities.

James J. Buckley, Esfandiar Eslami (2002)

Mathware and Soft Computing

We consider finite Markov chains where there are uncertainties in some of the transition probabilities. These uncertainties are modeled by fuzzy numbers. Using a restricted fuzzy matrix multiplication we investigate the properties of regular, and absorbing, fuzzy Markov chains and show that the basic properties of these classical Markov chains generalize to fuzzy Markov chains.

Graph fibrations, graph isomorphism, and PageRank

Paolo Boldi, Violetta Lonati, Massimo Santini, Sebastiano Vigna (2006)

RAIRO - Theoretical Informatics and Applications

PageRank is a ranking method that assigns scores to web pages using the limit distribution of a random walk on the web graph. A fibration of graphs is a morphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. We show that a deep connection relates fibrations and Markov chains with restart, a particular kind of Markov chains that include the PageRank one as a special case. This fact provides constraints on the...

Green functions on self-similar graphs and bounds for the spectrum of the laplacian

Bernhard Krön (2002)

Annales de l’institut Fourier

Combining the study of the simple random walk on graphs, generating functions (especially Green functions), complex dynamics and general complex analysis we introduce a new method for spectral analysis on self-similar graphs.First, for a rather general, axiomatically defined class of self-similar graphs a graph theoretic analogue to the Banach fixed point theorem is proved. The subsequent results hold for a subclass consisting of “symmetrically” self-similar graphs which however is still more general then...

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