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Markov Random Walk Representations with Continuous Distributions [article]

Chen-Hsiang Yeang, Martin Szummer
2012 arXiv   pre-print
Representations based on random walks can exploit discrete data distributions for clustering and classification. We extend such representations from discrete to continuous distributions.  ...  Transition probabilities are now calculated using a diffusion equation with a diffusion coefficient that inversely depends on the data density.  ...  random walks We begin by reviewing the discrete Markov random walk representation.  ... 
arXiv:1212.2510v1 fatcat:bqg4vqohorh27es6ihly4iwape

Page 2825 of Mathematical Reviews Vol. , Issue 84g [page]

1984 Mathematical Reviews  
In this paper the authors consider a restricted family of random walks, ¢.g., p(0,x)~dx'~* with 2<a<3, drop the assumption on the third moment, and obtain a representation for the G‘"’( x,y).  ...  This representation, with eigenvalue inequalities, leads to a characterization of the distribution of first hitting time as the convolution or mixture of distributions related to the geometric.  ... 

Page 4241 of Mathematical Reviews Vol. , Issue 88h [page]

1988 Mathematical Reviews  
Asymptotic representation of the potential of a lower continuous random walk on a Markov chain. (Russian) Analytic methods in reliability theory (Russian), 70-75, vi, Akad. Nauk Ukrain. SSR, Inst.  ...  Summary (translated from the Russian): “We obtain an asymptotic representation of the potential of a nondegenerate integer lower continuous random walk on a Markov chain in the case when the mathematical  ... 

Page 4829 of Mathematical Reviews Vol. , Issue 95h [page]

1995 Mathematical Reviews  
This paper deals with the derivation of exact joint distributions of random variables that are additive functionals defined on ex- cursions of a simple symmetric random walk.  ...  random walks.  ... 

Page 3980 of Mathematical Reviews Vol. , Issue 89G [page]

1989 Mathematical Reviews  
The distribution of the time and place of first passage over a level k for a left-continuous lattice random walk on a countable Markov chain is studied.  ...  It is proved under a mild condition that this Markov chain has at most one invariant distribution. As an application, random walks on R* with reflection at 0 are carefully studied.  ... 

Page 6334 of Mathematical Reviews Vol. , Issue 92k [page]

1992 Mathematical Reviews  
In conjunction with our discrete-time algorithm, this engenders an algorithm for construct- ing a Markov process representation for any distribution of con- tinuous phase-type.” 92k:60152 60310 60K25 Murthy  ...  “We also clarify the relation between the classes of continuous and discrete phase-type distributions, and show that O’Cinneide’s characterisation of continuous phase-type distributions is a corol- lary  ... 

Memory effects and static disorder reduce information in single-molecule signals [article]

Kevin Song, Dmitrii E Makarov, Etienne Vouga
2022 bioRxiv   pre-print
Toward this goal, a variety of low-dimensional models have been proposed as descriptions of single-molecule signals, including random walks with or without conformational memory and/or with static or dynamics  ...  ) and non-Markov single-molecule signals and between static and dynamic disorder.  ...  More generally, various anomalous-diffusion-type phenomena are often interpreted in terms of random walks on a network of "traps", with a certain distribution of the trap energies leading to a distribution  ... 
doi:10.1101/2022.01.13.476256 fatcat:36cij7c6cvffbcnzrzjoou676i

Page 4863 of Mathematical Reviews Vol. , Issue 99g [page]

1999 Mathematical Reviews  
with the help of a continuous product.”  ...  Let {S,, » > 1} be a random walk generated by i.i.d random vari- ables with ES; > 0.  ... 

Page 2659 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews  
We study a Markov process with discrete time and with a phase space formed by the continuous configurations.  ...  In this case the 60J Markov processes 994:60079 limit distribution of a Markov chain corresponds to the limit ‘Gibbs’ distribution on the set of continuous configurations.  ... 

Page 5137 of Mathematical Reviews Vol. , Issue 81M [page]

1981 Mathematical Reviews  
Authors’ summary: “The joint distributions, along with the gener- ating functions, of crossings of the line y=¢ and their runs are investigated for random walks corresponding to (i) samples of unequal  ...  {imit » for ie v.Y,) 0133 BERT ar PN HBB TA et BO 5137 Using the asymptotic representation of potentials of a lattice ran- dom walk, the author finds the limit distribution of the jump over the level x  ... 

Page 1407 of Mathematical Reviews Vol. 57, Issue 4 [page]

1979 Mathematical Reviews  
., the authors give necessary and sufficient conditions that the random walk have a limiting uniform distribution over the region.  ...  Nauk 1977, no. 4, 23-28, The author considers a k-dimensional random walk in a rectangu- lar region with reflecting, or partially reflecting walls.  ... 

Page 7326 of Mathematical Reviews Vol. , Issue 2001J [page]

2001 Mathematical Reviews  
Assuming that this random walk is nonsymmetric in the sense that Eln p # Eln(1 — p), the authors prove that M has a stationary distribution of the form r(x)du(x) i.e., abso- lutely continuous with respect  ...  (RS-AOS; Moscow) A random walk with an upper-continuous component, and the Lagrange inversion formula. (Russian. Russian summary) Teor. Veroyatnost.i Primenen. 45 (2000), no. 1, 166-175.  ... 

Page 1493 of Mathematical Reviews Vol. , Issue 88c [page]

1988 Mathematical Reviews  
Define a right random walk on S as the product Z, = X,X2---Xn, n > 1, where the factors are independent and identically distributed with law yw.  ...  Let S = (So, Si,---) be a random walk with S; in the domain of attraction of a stable distribution. Let S and S be two independent copies of S. Define M, = maxo<j<; 5S; and my = Maxg<j<;(—5Si).  ... 

Page 4113 of Mathematical Reviews Vol. , Issue 99f [page]

1999 Mathematical Reviews  
Compactness and limit properties for the random walk model; 6. Laplace principle for the random walk model with continuous statistics; 7.  ...  Extension of the Laplace principle for the empirical measures of a Markov chain; 10. Laplace principle for the continuous time Markov processes with continuous statistics.  ... 

The cutoff phenomenon in finite Markov chains

P. Diaconis
1996 Proceedings of the National Academy of Sciences of the United States of America  
It shows that chains with polynomial growth (drunkard's walk) do not show cutoffs.  ...  Natural mixing processes modeled by Markov chains often show a sharp cutoff in their convergence to long-time behavior.  ...  These heuristics thus predict a cutoff for random walks where the size of the representations grow with the size of the group.  ... 
doi:10.1073/pnas.93.4.1659 pmid:11607633 pmcid:PMC39998 fatcat:kxswbudqfvetbohxy6ailuv4t4
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