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Keywords
(9)
Branching Random Walk
hausdorff dimension
Indexation
Packing Dimension
Spectral Radius
Transition Matrix
Transition Probability
Tree Structure
Markov Chain
Related Publications
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Markov Chains Indexed by Trees
Markov Chains Indexed by Trees,10.1214/aop/1176988857,Annals of Probability,Itai Benjamini,Yuval Peres
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Markov Chains Indexed by Trees
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Citations: 48
)
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Itai Benjamini
,
Yuval Peres
We study a variant of branching Markov chains in which the branching is governed by a fixed deterministic tree $T$ rather than a GaltonWatson process. Sample path properties of these chains are determined by an interplay of the
tree structure
and the transition probabilities. For instance, there exists an infinite path in $T$ with a bounded trajectory iff the
Hausdorff dimension
of $T$ is greater than $\log(1/\rho)$ where $\rho$ is the
spectral radius
of the transition matrix.
Journal:
Annals of Probability  ANN PROBAB
, vol. 22, no. 1994, pp. 219243, 1994
DOI:
10.1214/aop/1176988857
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Citation Context
(28)
...The notion of a treeindexed random walk was rst introduced and studied by Benjamini and Peres [
4
]...
Benny Sudakov
,
et al.
A randomized embedding algorithm for trees
...Athreya and Kang [4], Benjamini and Peres [
8
]), where for every x, F(1)(x, θ) and F(2)(x, θ) are i.i.d...
...The law of large numbers that we obtain is a continuous time version of the law of large numbers in Benjamini and Peres [
8
], Delmas and Marsalle [17], with possible asymmetric branching and random number of offspring...
Vincent Bansaye
,
et al.
Limit theorems for Markov processes indexed by continuous time Galton...
...This subject has been studied in the literature (see e.g., [6,
8
]) in the symmetric independent case...
Vincent Bansaye
.
Proliferating parasites in dividing cells: Kimmel’s branching model re...
...Definition 1 [
1
] . Let T be an infinite tree, S be a finite state space, {Xt ,t ∈ T } be a collection...
...The subject of treeindexed processes is rather young. Benjamini and Peres [
1
] have given...
Huilin Huang
,
et al.
Strong law of large numbers for Markov chains indexed by an infinite t...
...Now, there are three possible regimes for irreducible BMC: transient ( (x) = 0 8x), weakly recurrent (0 < (x) < 1 for some x) and strongly recurrent ( (x) = 1 8x), compare with Gantert and M¨uller [4] and Benjamini and Peres [
1
]...
...We define recurrence and transience for BMC in analogy to [
1
] and [4]:...
...In fact, due to the irreducibility, (x) > 0 and (x) = 0 hold either for all or none x 2 X. This can be shown analogously to Lemma 3.1 in [
1
]...
Sebastian Muller
.
A criterion for transience of multidimensional branching random walk i...
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Citations
(48)
Inhibition of human drug metabolizing cytochrome P450 enzymes by plant isoquinoline alkaloids
(
Citations: 2
)
Kaisa A. Salminen
,
Achim Meyer
,
Lenka Jerabkova
,
Laura E. Korhonen
,
Minna Rahnasto
,
Risto O. Juvonen
,
Peter Imming
,
Hannu Raunio
Journal:
Phytomedicine
, vol. 18, no. 6, pp. 533538, 2011
The Asymptotic Equipartition Property for Homogeneous Markov Chains Indexed by Trees
Weicai Peng
Conference:
International Conference on Business Computing and Global Informatization  BCGIN
, 2011
A randomized embedding algorithm for trees
(
Citations: 2
)
Benny Sudakov
,
Jan Vondrák
Journal:
Combinatorica
, vol. 30, no. 4, pp. 445470, 2010
Branching Feller diffusion for cell division with parasite infection
(
Citations: 2
)
Vincent BansayeViet
,
Viet Chi Tran
Published in 2010.
Some Limit Properties for Nonhomogeneous Tree Indexed Markov Chains
Fan Zhenyao
,
Zhai Quanli
,
Jin Shaohua
Conference:
ACIS International Symposium on Cryptography and Network Security, Data Mining and Knowledge Discovery, ECommerce & Its Applications and Embedded Systems  CDEE
, 2010