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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 364 (33 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...
Limiting behaviour of sums and the term of maximum modulus
 Proc. Lond. Math. Soc
, 1984
"... Suppose that {Xn: n ^ 1} are independent and identically distributed random variables with common continuous distribution function F. Set Sn = Xt +... + Xn and Mn = V" * i, and let X[l) be the term of maximum modulus, i.e. the Xt among Xu...,Xn for which  Xt  is largest. The influence of the ..."
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Cited by 9 (6 self)
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Suppose that {Xn: n ^ 1} are independent and identically distributed random variables with common continuous distribution function F. Set Sn = Xt +... + Xn and Mn = V" * i, and let X[l) be the term of maximum modulus, i.e. the Xt among Xu...,Xn for which  Xt  is largest. The influence of the extreme terms on the sample sum is studied by examining the behaviour of Sn/X[l) and Sn/Mn. The main results centre about conditions for these quantities to converge to 1 in probability and almost surely. Related results deal with ratios of order statistics and ratios of record values of {Xn}. A novel feature of our approach is to study the behaviour of {SB} between successive record values of {\Xn\}. 1. Introduction and
Convergence of point processes with weakly dependent points
, 2008
"... For each n ≥ 1, let {Xj,n}1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process Nn = ∑n j=1 δXj,n to an infinitely divisible point process. From the point process conve ..."
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Cited by 8 (2 self)
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For each n ≥ 1, let {Xj,n}1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process Nn = ∑n j=1 δXj,n to an infinitely divisible point process. From the point process convergence, we obtain the convergence in distribution of the partial sum sequence Sn = ∑n Xj,n to an infinitely divisible random j=1 variable, whose Lévy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (rowwise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model.
AGING IN REVERSIBLE DYNAMICS OF DISORDERED SYSTEMS. II. emergence of the arcsine . . .
, 2010
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LévyLepage Series Representation of Stable Vectors. Convergence in Variation
, 1999
"... Multidimensional stable laws G# admit a wellknown LevyLePage series representation G# = L # # # j=1 # 1/# j X j # , 0 < # < 2, where #1 , #2 , . . . are the successive times of jumps of a standard Poisson process, and X1 , X2 , . . . denote i.i.d. random vectors, independent of # ..."
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Cited by 7 (1 self)
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Multidimensional stable laws G# admit a wellknown LevyLePage series representation G# = L # # # j=1 # 1/# j X j # , 0 < # < 2, where #1 , #2 , . . . are the successive times of jumps of a standard Poisson process, and X1 , X2 , . . . denote i.i.d. random vectors, independent of #1 , #2 , . . . We present (asymptotically) optimal bounds for the total variation distance between a stable law and the distribution of a partial sum of the LévyLePage series. In the onedimensional case similar results were obtained earlier by Bentkus, Götze and Paulauskas.