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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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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...
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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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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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
Extreme Value Theory for Random Exponentials
"... Abstract. We study the limit distribution of upper extreme values of i.i.d. exponential samples {etXi} N i=1 as t → ∞, N → ∞. Two cases are considered: (A) ess sup X = 0 and (B) ess sup X = ∞. We assume that the function h(x) = − log P{X> x} (case B) or h(x) = − log P{X> −1/x} (case A) is (norma ..."
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Abstract. We study the limit distribution of upper extreme values of i.i.d. exponential samples {etXi} N i=1 as t → ∞, N → ∞. Two cases are considered: (A) ess sup X = 0 and (B) ess sup X = ∞. We assume that the function h(x) = − log P{X> x} (case B) or h(x) = − log P{X> −1/x} (case A) is (normalized) regularly varying at ∞ with index 1 < ϱ < ∞ (case B) or 0 < ϱ < ∞ (case A). The growth scale of N is chosen in the form N ∼ eλH0(t) (0 < λ < ∞), where H0(t) is a certain asymptotic version of the function H(t): = log E[etX] (case B) or H(t) = − log E[etX] (case A). As shown earlier by Ben Arous et al. [5], there are critical points λ1 < λ2, below which the LLN and CLT, respectively, break down, whereas for 0 < λ < λ2 the limit laws for the sum SN (t) = etX1 + · · · + etXN prove to be stable, with characteristic exponent α = α(ϱ, λ) ∈ (0, 2). In this paper, we obtain the (joint) limit distribution of the upper order statistics of the exponential sample. In particular, M1,N = max{etXi} N i=1 has asymptotically the Fréchet distribution with parameter α. We also show that the empirical extremal measure converges (in fdd) to a Poisson random measure with intensity d(x−α). These results are complemented by explicit representations of the joint limit distribution of SN (t) and M1,N (t) (and in particular of their ratio) in terms of i.i.d. random variables with standard exponential distribution. 1.
The Ratio of the Extreme to the Sum in a Random Sequence with Applications
, 1994
"... If X 1 , X 2 , . . . , X n is a sequence of nonnegative independent random variables with common distribution function F (t), we write X (n) for the maximum of the sequence and S n for its sum. The ratio variate R n = X (n) /S n is a quantity arising in the analysis of process speedup and the perf ..."
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If X 1 , X 2 , . . . , X n is a sequence of nonnegative independent random variables with common distribution function F (t), we write X (n) for the maximum of the sequence and S n for its sum. The ratio variate R n = X (n) /S n is a quantity arising in the analysis of process speedup and the performance of scheduling. O'Brien (1980) showed that R n ® 0 almost surely Û EX 1 < as n ® . Since {R n } is a uniformly bounded sequence it follows that EX 1 < Þ ER n ® 0 as n ® . Here we show that, provided either that (i) EX 1 2 < or that (ii) 1  F(t) is a regularly varying function with index r < 1, it follows that ER n = ES n EX (n) ###### # # 1 + o(1) # # (n ® ) . Since the asymptotics of EX (n) is often readily calculated, this provides a useful estimate for the most significant behavior of the ratio R n in expectation. We apply this result to multiprocessor scheduling policies and to the behavior of sample statistics. June 28, 1994 Department of Computer Science The Univer...
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...
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.”