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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...
On the distribution of ranked heights of excursions of a Brownian bridge
 In preparation
, 1999
"... The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest maximum over a positive excursion of the bridge has the same distribution as M br+ 1 =j, where th ..."
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Cited by 11 (6 self)
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The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest maximum over a positive excursion of the bridge has the same distribution as M br+ 1 =j, where the distribution of M br+ 1 = sup 0t1 B br t is given by L'evy's formula P (M br+ 1 ? x) = e \Gamma2x 2 . The probability density of the height M br j of the jth highest maximum of excursions of the reflecting Brownian bridge (jB br t j; 0 t 1) is given by a modification of the known `function series for the density of M br 1 = sup 0t1 jB br t j. These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized bridge of a selfsimilar recurrent Markov process. Keywords: Brownian bridge, Brownian excursion, Brownian scaling, local time, selfsimilar recurrent Markov process, Bessel p...
The twoparameter PoissonDirichlet point process
, 2007
"... The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtai ..."
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Cited by 3 (0 self)
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The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Relying on this, we apply the theory of point processes to reveal mathematical structure of the twoparameter PoissonDirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we will be able to extend several results previously known for the oneparameter case, and the MarkovKrein identity for the generalized Dirichlet process is discussed from a point of view of functional analysis based on the twoparameter PoissonDirichlet distribution. 1
Limit theorems for sums of order statistics
 In: Sixth Interna CSORGO, HAEUSLER AND MASON tionaI Summer School in Probability theory and Mathematical Statistics
, 1988
"... This is a brief summary of recent results on the asymptotic distribution of various ordered portions of sums of independent, identically distributed random variables that have been obtained by a direct probabilistic approach based upon some integral representations of such sums in terms of uniform e ..."
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Cited by 1 (0 self)
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This is a brief summary of recent results on the asymptotic distribution of various ordered portions of sums of independent, identically distributed random variables that have been obtained by a direct probabilistic approach based upon some integral representations of such sums in terms of uniform empirical distribution functions, following a quantile transformation. Almost all the results discussed here have been obtained jointly with Erich Haeusler and David M. Mason. The survey itself forms a guideline for a series of lectures given as a part of the Sixth International Summer School. 1.
A Higher Order Expansion for the Joint Density of the Sum and the Maximum with Applications to the Estimation of Climatological Trends
, 2000
"... The higher order expansion for the joint density of the sum and maximum of an iid sequence answers two questions: the theoretical question of what is the rate of the asymptotic independence between these two which was established by Chow and Teugels (1978) and Anderson and Turkman (1991), and the ..."
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Cited by 1 (0 self)
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The higher order expansion for the joint density of the sum and maximum of an iid sequence answers two questions: the theoretical question of what is the rate of the asymptotic independence between these two which was established by Chow and Teugels (1978) and Anderson and Turkman (1991), and the practical question of how to describe and model the dependence when the asymptotic result is not yet realized. Developing such an expansion under the three different domains of attraction for the maximum and modeling the annual total and maximum precipitation across the contiguous US, using a combined generalized extreme value (GEV) version of the expansion, are the key elements of this thesis. The three
Limit Theorems on Ordered Random Vectors
"... Let X 1 ; X 2 ; ::: be independent identically distributed sdimensional random vectors, whose distribution belongs to the domain of attraction of a stable law. Let's X j;n ; j = 1; 2; ; n denote the order statistics built by increase of norms of random vectors X 1 ; X 2 ; ; X n ; i.e. jX 1;n ..."
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Let X 1 ; X 2 ; ::: be independent identically distributed sdimensional random vectors, whose distribution belongs to the domain of attraction of a stable law. Let's X j;n ; j = 1; 2; ; n denote the order statistics built by increase of norms of random vectors X 1 ; X 2 ; ; X n ; i.e. jX 1;n j jX 2;n j jX n;n j: We investigate the asymptotic properties of random vectors T nk = (X 1n + +X n k;n )=jX n k+1;n j. 1 Introduction Let X 1 ; X 2 ; :::; X n ; ::: be independent identically distributed (i.i.d.) sdimensional random vectors having common absolute continuous distribution function. Let S(n) = X 1 +X 2 + ::: + X n ; F (x) = P (jX 1 j x); and p(x) be density of distribution of vector X 1 . sdimensional distribution is called to be stable if to every pair of vectors A 1 and A 2 and positive numbers B 1 and B 2 there always correspond a vector A and a positive number B such that for the three independent random vectors X, X 1 and X 2 possessing thi...
Professor Zipf goes to Wall Street ∗
, 2009
"... The heavytailed distribution of firm sizes first discovered by Zipf (1949) is one of the best established empirical facts in economics. We show that it has strong implications for asset pricing. Due to the concentration of the market portfolio when the distribution of the capitalization of firms is ..."
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The heavytailed distribution of firm sizes first discovered by Zipf (1949) is one of the best established empirical facts in economics. We show that it has strong implications for asset pricing. Due to the concentration of the market portfolio when the distribution of the capitalization of firms is sufficiently heavytailed, an additional risk factor generically appears even for very large economies. Our twofactor model is as successful empirically as the threefactor FamaFrench model. The authors acknowledge helpful discussions and exchanges with Marco Avellaneda, Emanuele Bajo,
The asymptotic distribution of randomly weighted sums and selfnormalized sums ∗
"... E l e c t r o n ..."
Wireless Link Quality Modelling and Mobility Management for Cellular Networks Reviewers:
, 2012
"... framework of Conventions Industrielles de Formation par la REcherche (CIFRE) ..."
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framework of Conventions Industrielles de Formation par la REcherche (CIFRE)