Doob–Martin boundary of Rémy’s tree growth chain

Evans, Steven N.; Grübel, Rudolf; Wakolbinger, Anton

doi:10.1214/16-aop1112

Cited by 16 publications

(58 citation statements)

References 31 publications

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“…The backward transition probabilities of the Rémy chain were identified in [7] and they coincide with those of the PATRICIA chains. It follows that the common family of infinite bridges for the PATRICIA chains is the same as that of the Rémy chain and the latter was determined in [7]. An outline of the remainder of the paper is as follows.…”

Section: Introductionmentioning

confidence: 72%

“…We stress that this gap does not illegitimate the further development in [7] which does not depend on the details of the definition of a didendritic system but only on the two facts that a finite didendritic system is effectively a finite leaf-labeled binary tree and that the class of didentritic systems is closed under projective limits in a natural way -both of which are true for our new definition. We also develop in an explicit manner an alternative description of the class of finite didendritic systems that was somewhat implicit in [7]. This alternative description, which is obtained in Proposition 6.7, is crucial for the later exposition.…”

Section: Introductionmentioning

confidence: 93%

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PATRICIA Bridges

Evans

Wakolbinger

2020

Genealogies of Interacting Particle Systems

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A radix sort tree arises when storing distinct infinite binary words in the leaves of a binary tree such that for any two words their common prefixes coincide with the common prefixes of the corresponding two leaves. If one deletes the out-degree 1 vertices in the radix sort tree and "closes up the gaps", then the resulting PATRICIA tree maintains all the information that is necessary for sorting the infinite words into lexicographic order.We investigate the PATRICIA chains -the tree-valued Markov chains that arise when successively building the PATRICIA trees for the collection of infinite binary words Z 1 , . . . , Zn, n = 1, 2, . . ., where the source words Z 1 , Z 2 , . . . are independent and have a common diffuse distribution on {0, 1} ∞ . It turns out that the PATRICIA chains share a common collection of backward transition probabilities and that these are the same as those of a chain introduced by Rémy for successively generating uniform random binary trees with larger and larger numbers of leaves. This means that the infinite bridges of any PATRICIA chain (that is, the chains obtained by conditioning a PATRICIA chain on its remote future) coincide with the infinite bridges of the Rémy chain. The infinite bridges of the Rémy chain are characterized concretely in Evans, Grübel, and Wakolbinger 2017 and we recall that characterization here while adding some details and clarifications.

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Section: Introductionmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 93%

PATRICIA Bridges

Evans

Wakolbinger

2020

Genealogies of Interacting Particle Systems

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“…The following result can be established using essentially the same argument as in Proposition 5.19 (see also the subsequent Remark 5.20) of [EGW15], and we omit the details.…”

Section: Characterization Of Exchangeable Random Total Ordersmentioning

confidence: 96%

“…This is, in turn, equivalent to showing that the corresponding labeled infinite bridge induces an ergodic exchangeable random order. The latter, however, can be established along the lines of [EGW15, Corollary 5.21] and [EW16, Corollary 7.2], so we omit the details.…”

Section: Characterization Of Exchangeable Random Total Ordersmentioning

confidence: 99%

Doob–Martin compactification of a Markov chain for growing random words sequentially

Choi

Evans

2017

Stochastic Processes and their Applications

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We consider a Markov chain that iteratively generates a sequence of random finite words in such a way that the nth word is uniformly distributed over the set of words of length 2n in which n letters are a and n letters are b: at each step an a and a b are shuffled in uniformly at random among the letters of the current word. We obtain a concrete characterization of the Doob-Martin boundary of this Markov chain and thereby delineate all the ways in which the Markov chain can be conditioned to behave at large times. Writing N(u) for the number of letters a (equivalently, b) in the finite word u, we show that a sequence (un)n∈ℕ of finite words converges to a point in the boundary if, for an arbitrary word ν, there is convergence as n tends to infinity of the probability that the selection of N(ν) letters a and N(ν) letters b uniformly at random from un and maintaining their relative order results in ν. We exhibit a bijective correspondence between the points in the boundary and ergodic random total orders on the set {a1, b1, a2, b2, …} that have distributions which are separately invariant under finite permutations of the indices of the a's and those of the b's. We establish a further bijective correspondence between the set of such random total orders and the set of pairs (μ, ν) of diffuse probability measures on [0,1] such that ½(μ + ν) is Lebesgue measure: the restriction of the random total order to {a1, b1,…, an, bn} is obtained by taking X1,…, Xn (resp. Y1,… ,Yn) i.i.d. with common distribution μ (resp. ν), letting (Z1,…, Z2n) be {X1, Y1,…, Xn, Yn} in increasing order, and declaring that the kth smallest element in the restricted total order is ai (resp. bj) if Zk = Xi (resp. Zk = Yj).

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“…We end this sketch with a disclaimer: The beauty and elegance of the general theory notwithstanding, its actual implementation for a specific Markov chain, such as a description of the boundary and the conditioned chains, may be far from being trivial; see [EGW16] for a recent example. Independent of their applications in the context of the election algorithm it may therefore be of interest that the two chains introduced in Section 1 permit a comparably short and explicit treatment along the above lines.…”

Section: Boundary Theory For Space-time Markov Chainsmentioning

confidence: 99%

Leader election: a Markov chain approach

Grübel

Hagemann

2016

MathAppl

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ABSTRACT. A well-studied randomized election algorithm proceeds as follows: In each round the remaining candidates each toss a coin and leave the competition if they obtain heads. Of interest is the number of rounds required and the number of winners, both related to maxima of geometric random samples, as well as the number of remaining participants as a function of the number of rounds. We introduce two related Markov chains and use ideas and methods from discrete potential theory to analyse the respective asymptotic behaviour as the initial number of participants grows. One of the tools used is the approach via the Rényi-Sukhatme representation of exponential order statistics, which was first used in the leader election context by Bruss and Grübel in [BG03].

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Doob–Martin boundary of Rémy’s tree growth chain

Cited by 16 publications

References 31 publications

PATRICIA Bridges

PATRICIA Bridges

Doob–Martin compactification of a Markov chain for growing random words sequentially

Leader election: a Markov chain approach

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