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Random walk centrality and a partition of Kemeny's constant

Stephen J. Kirkland (2016)

Czechoslovak Mathematical Journal

We consider an accessibility index for the states of a discrete-time, ergodic, homogeneous Markov chain on a finite state space; this index is naturally associated with the random walk centrality introduced by Noh and Reiger (2004) for a random walk on a connected graph. We observe that the vector of accessibility indices provides a partition of Kemeny's constant for the Markov chain. We provide three characterizations of this accessibility index: one in terms of the first return time to the state...

Random walk in random environment with asymptotically zero perturbation

M.V. Menshikov, Andrew R. Wade (2006)

Journal of the European Mathematical Society

We give criteria for ergodicity, transience and null-recurrence for the random walk in random environment on + = { 0 , 1 , 2 , } , with reflection at the origin, where the random environment is subject to a vanishing perturbation. Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different from the previously studied cases. Our method is based on a martingale technique—the method of Lyapunov functions.

Random walk local time approximated by a brownian sheet combined with an independent brownian motion

Endre Csáki, Miklós Csörgő, Antónia Földes, Pál Révész (2009)

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

Let ξ(k, n) be the local time of a simple symmetric random walk on the line. We give a strong approximation of the centered local time process ξ(k, n)−ξ(0, n) in terms of a brownian sheet and an independent Wiener process (brownian motion), time changed by an independent brownian local time. Some related results and consequences are also established.

Random walk on a building of type Ãr and brownian motion of the Weyl chamber

Bruno Schapira (2009)

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

In this paper we study a random walk on an affine building of type Ãr, whose radial part, when suitably normalized, converges toward the brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane (Probab. Theory Related Fields89 (1991) 117–129). This extends also the link at the probabilistic level between riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol...

Random walk on graphs with regular resistance and volume growth

András Telcs (2008)

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

In this paper characterizations of graphs satisfying heat kernel estimates for a wide class of space–time scaling functions are given. The equivalence of the two-sided heat kernel estimate and the parabolic Harnack inequality is also shown via the equivalence of the upper (lower) heat kernel estimate to the parabolic mean value (and super mean value) inequality.

Random Walks and Trees

Zhan Shi (2011)

ESAIM: Proceedings

These notes provide an elementary and self-contained introduction to branching random walks. Section 1 gives a brief overview of Galton–Watson trees, whereas Section 2 presents the classical law of large numbers for branching random walks. These two short sections are not exactly indispensable, but they introduce the idea of using size-biased trees, thus giving motivations and an avant-goût to the main part, Section 3, where branching random walks...

Random walks on co-compact fuchsian groups

Sébastien Gouëzel, Steven P. Lalley (2013)

Annales scientifiques de l'École Normale Supérieure

It is proved that the Green’s function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence R . It is also shown that Ancona’s inequalities extend to  R , and therefore that the Martin boundary for  R -potentials coincides with the natural geometric boundary S 1 , and that the Martin kernel is uniformly Hölder continuous. Finally, this implies a local limit theorem for the transition probabilities: in the aperiodic case, p n ( x , y ) C x , y R - n n - 3 / 2 .

Ranked fragmentations

Julien Berestycki (2002)

ESAIM: Probability and Statistics

In this paper we define and study self-similar ranked fragmentations. We first show that any ranked fragmentation is the image of some partition-valued fragmentation, and that there is in fact a one-to-one correspondence between the laws of these two types of fragmentations. We then give an explicit construction of homogeneous ranked fragmentations in terms of Poisson point processes. Finally we use this construction and classical results on records of Poisson point processes to study the small-time...

Ranked Fragmentations

Julien Berestycki (2010)

ESAIM: Probability and Statistics

In this paper we define and study self-similar ranked fragmentations. We first show that any ranked fragmentation is the image of some partition-valued fragmentation, and that there is in fact a one-to-one correspondence between the laws of these two types of fragmentations. We then give an explicit construction of homogeneous ranked fragmentations in terms of Poisson point processes. Finally we use this construction and classical results on records of Poisson point processes to study the small-time behavior...

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