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The twoparameter PoissonDirichlet distribution derived from a stable subordinator.
, 1995
"... The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov ..."
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Cited by 221 (37 self)
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The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to VershikShmidtIgnatov, are generalized to the twoparameter case. The sizebiased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
PoissonKingman Partitions
 of Lecture NotesMonograph Series
, 2002
"... This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a subordin ..."
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Cited by 11 (3 self)
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This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a subordinator, that is an increasing process with stationary independent increments. Examples include the twoparameter family of PoissonDirichlet models derived from the Poisson process of jumps of a stable subordinator. Applications are made to the random partition generated by the lengths of excursions of a Brownian motion or Brownian bridge conditioned on its local time at zero.
M.: Universality classes for extreme value statistics
 Journal of Physics A
, 1997
"... The low temperature physics of disordered systems is governed by the statistics of extremely low energy states. It is thus rather important to discuss the possible universality classes for extreme value statistics. We compare the usual probabilistic classification to the results of the replica appro ..."
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Cited by 9 (0 self)
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The low temperature physics of disordered systems is governed by the statistics of extremely low energy states. It is thus rather important to discuss the possible universality classes for extreme value statistics. We compare the usual probabilistic classification to the results of the replica approach. We show in detail that one class of independent variables corresponds exactly to the socalled one step replica symmetry breaking solution in the replica language. This universality class holds if the correlations are sufficiently weak. We discuss the relation between the statistics of extremes and the problem of Burgers turbulence in decay. LPTENS preprint 97/XX
Bouchaud: Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
 J. Phys. A: Math. Theor
, 2008
"... potential ..."
Universality Classes For Extreme Value Statistics
"... The low temperature physics of disordered systems is governed by the statistics of extremely low energy states. It is thus rather important to discuss the possible universality classes for extreme value statistics. We compare the usual probabilistic classification to the results of the replica appro ..."
Abstract
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The low temperature physics of disordered systems is governed by the statistics of extremely low energy states. It is thus rather important to discuss the possible universality classes for extreme value statistics. We compare the usual probabilistic classification to the results of the replica approach. We show in detail that one class of independent variables corresponds exactly to the socalled one step replica symmetry breaking solution in the replica language. This universality class holds if the correlations are sufficiently weak. We discuss the relation between the statistics of extremes and the problem of Burgers turbulence in decay. LPTENS preprint 97/XX Electronic addresses : bouchaud@amoco.saclay.cea.fr, mezard@physique.ens.fr 1 Introduction The replica method is one of the very few general analytical methods to investigate disordered systems [1]. Although the physical meaning of Parisi's `replica symmetry breaking' (rsb) scheme needed to obtain the correct low temperature s...
Keywords: Limiting Behavior of Coefficients of Variation, Nonself Averaging Behavior, MittagLeffler Distributions
, 2006
"... CIRJE Discussion Papers can be downloaded without charge from: ..."
LOCALIZATION VS. DELOCALIZATION OF RANDOM DISCRETE MEASURES*
"... Abstract. Sequences of discrete measures µ(n) with random atoms {µ (n) i, i =1, 2,...} such that i µ(n) i = 1are considered. The notions of (complete) asymptotic localization vs. delocalization of such measures in the weak (mean or probability) and strong (with probability 1) sense are proposed and ..."
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Abstract. Sequences of discrete measures µ(n) with random atoms {µ (n) i, i =1, 2,...} such that i µ(n) i = 1are considered. The notions of (complete) asymptotic localization vs. delocalization of such measures in the weak (mean or probability) and strong (with probability 1) sense are proposed and analyzed, proceeding from the standpoint of the largest atoms ’ behavior as n →∞. In this framework, the class of measures with the atoms of the form µ (n) i = Xi/Sn (i =1,...,n) is studied, where X1,X2,... is a sequence of positive, independent, identically distributed random variables (with a common distribution function F) and Sn = X1 + ···+ Xn. If E[X1] < ∞, then the law of large numbers implies that µ(n) is strongly delocalized. The case where E[X1] = ∞ is studied under the standard assumption that F has a regularly varying upper tail (with exponent 0 ≦ α ≦ 1). It is shown that for α<1, weak localization occurs. In the critical point α = 1, the weak delocalization is established. For α = 0, localization is strong unless the tail decay is “hardly slow.”