## Random Discrete Distributions Derived From Self-Similar Random Sets (1996)

Venue: | Electronic J. Probability |

Citations: | 15 - 10 self |

### BibTeX

@ARTICLE{Pitman96randomdiscrete,

author = {Jim Pitman and Marc Yor},

title = {Random Discrete Distributions Derived From Self-Similar Random Sets},

journal = {Electronic J. Probability},

year = {1996},

volume = {1},

pages = {1--28}

}

### OpenURL

### Abstract

: A model is proposed for a decreasing sequence of random variables (V 1 ; V 2 ; \Delta \Delta \Delta) with P n V n = 1, which generalizes the Poisson-Dirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let V n be the length of the nth longest component interval of [0; 1]nZ, where Z is an a.s. non-empty random closed of (0; 1) of Lebesgue measure 0, and Z is self-similar, i.e. cZ has the same distribution as Z for every c ? 0. Then for 0 a ! b 1 the expected number of n's such that V n 2 (a; b) equals R b a v \Gamma1 F (dv) where the structural distribution F is identical to the distribution of 1 \Gamma sup(Z " [0; 1]). Then F (dv) = f(v)dv where (1 \Gamma v)f(v) is a decreasing function of v, and every such probability distribution F on [0; 1] can arise from this construction. Keywords: interval partition, zero set, excursion lengths, regenerative set, structural distribution. AMS subject classificat...

### Citations

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Citation Context ...subsets of R I . A real or vector-valued process (X t ; ts0) is called fi-self-similar where fi 2 R I if for every c ? 0 (X ct ; ts0) d = (c fi X t ; ts0) (20) Such processes were studied by Lamperti =-=[24, 25]-=-, who called them semistable. See [40] for a survey of the literature of these processes. A random closed subset Z of R I is self-similar in the sense (6) iff its age process is 1-selfsimilar. A natur... |

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Citation Context ... \Delta \Delta \Delta ) (1) This construction, and its iteration to define a size-biased random permutationsof (V n ), play a key role in both theory and applications of random discrete distributions =-=[14, 8, 33]-=-. Denote by F the distribution on (0; 1] of a size-biased pick V from (V n ). Following Engen [10], call F the structural distribution of (V n ). It is well known that many probabilities and expectati... |

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Some limit theorems for sums of independent random variables with infinite mathematical expectations
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Citation Context ...rom (V n ), as in the previous example. So the structural distribution of (V n ) is again identical to the distribution of A 1 , in this case beta (1 \Gamma ff; ff), also known as generalized arcsine =-=[9]-=-. It was shown further in [30] that in this example a size-biased random permutation ( ~ V n ) of (V n ), constructed with extra randomization, admits the representation (12) for independent beta (1 \... |

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Citation Context ...n, and ff = (2 \Gamma ffi)=2 for a Bessel process of dimension ffi. The distribution of (V n ) in this case is an analog of the Poisson-Dirichlet distribution that has been studied by several authors =-=[44, 29, 35]-=-. It is well known that this Z is a.s. perfect, i.e. Z contains no isolated points. Consequently, Z is uncountable, and its points cannot be simply ranked as in the previous example. Still, it was sho... |

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- 1996
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Citation Context ...mega Q and Q\Omega P will be three distinct laws for a rdd with the same structural distribution str(P)sstr(Q). A nice illustration of the composition operation is provided by the following result of =-=[34]-=-: for ff ? 0 and ` ? 0, pd(0; `)\Omega pd(ff; 0) = pd(ff; `) (67) If, as in the above examples, both P and Q can be derived from a self-sim 0 set, it is natural to ask whether P ffl Q and P\Omega Q ca... |

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Citation Context ...mple 13 stationary-regen 0 (F 0 ). This is the unique distribution of a stationary 0 set Z that is regenerative subset of R I in the sense of [26], and such that L has distribution F 0 . See Fristedt =-=[13]-=-, who gives the following construction of Z, and further references. Let Z = f\GammaA 0 \Gamma Z 1 g [ fD 0 + Z 2 g (35) where A 0 and D 0 are defined by (34) in terms of L with distribution F 0 and a... |

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10 |
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Citation Context ... We provide another approach to this result in Section 4. The idea is to exploit the fact that Z is self-similar iff log Z is stationary, and make use of the generalizations to stationary random sets =-=[31]-=- of some standard formulae for stationary renewal processes. An advantage of this approach is that it gives an explicit description of all possible joint distributions of (G t ; D t ) derived from a s... |

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8 |
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Citation Context ...] at the points of Z. One natural construction of such a Z, corresponding to an arbitrary prescribed distribution for (V n ), is obtained from the exchangeable interval partition considered by Berbee =-=[3]-=- and Kallenberg [17]. Here we consider constructions with a different sort of symmetry: Definition 3 self-sim 0 set. Call Z self-similar if Z d = cZ for all c ? 0; (6) where cZ = fcz; z 2 Zg, and d = ... |

8 |
Some extensions of the arc sine law as partial consequences of the scaling property of Brownian motion
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Citation Context ... The zero set of a perturbed Brownian motion. The above formulae can be applied to the self-sim 0 set Z defined by the zero set of the perturbed Brownian motion (jB t j \Gamma ��L t ; ts0) studied=-= in [5]. He-=-re B is a standard Brownian motion, (L t ; ts0) is its local time process at zero, and �� ? 0 is a parameter. The law of G 1 , found explicitly in [5] turns out to be fairly complicated. Still, wi... |

8 |
Stochastic abundance models with emphasis on biological communities and species
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Citation Context ...s [12, 1], ffl asymptotic distributions in combinatorics and number theory [42, 43, 41], ffl models for gene frequencies in population genetics [20, 6, 11] ffl models for species diversity in ecology =-=[28, 10]-=-, ffl the representation of partition structures [21], ffl models for storage and search [4, 19, 7] ffl analysis of the zero sets of stochastic processes such as Brownian motion and Brownian bridge [4... |

7 |
Splitting at backward times in regenerative sets
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Citation Context ...hat derived from the stable(ff) set Z, as in Example 6, while pd(ff; ff) is the distribution of (V n ) derived from this stable (ff) set Z by conditioning on 0 2 Z, an operation made precise in [44], =-=[18]-=-. A sequences(V n ) with pd(ff; `)distribution can be constructed by ranking ( ~ V n ) defined by the residual allocation model (12) for independent X n such that X n has beta (1 \Gamma ff; `+nff) dis... |

5 |
Size-biased filtering of Poisson-Dirichlet samples with an application to partition structures in genetics
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Citation Context ... \Delta \Delta \Delta ) (1) This construction, and its iteration to define a size-biased random permutationsof (V n ), play a key role in both theory and applications of random discrete distributions =-=[14, 8, 33]-=-. Denote by F the distribution on (0; 1] of a size-biased pick V from (V n ). Following Engen [10], call F the structural distribution of (V n ). It is well known that many probabilities and expectati... |

4 | Sur les ferm'es al'eatoires - Az'ema - 1985 |

4 |
On a constant arising in the theory of symmetric groups and on PoissonDirichlet measures. Theory Probab
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(Show Context)
Citation Context ...\Delta \Delta \Deltag (10) It is known that Z n may be represented as Z n = (1 \Gamma X 1 ) \Delta \Delta \Delta (1 \Gamma X n ) (ns1) (11) where X 1 = A 1 and the X i are i.i.d. beta(1; `) variables =-=[16]-=-. In terms of the X i the sequence (V n ) is obtained by ranking the terms ~ V n defined by ~ V 1 = X 1 ; ~ V n = (1 \Gamma X 1 ) \Delta \Delta \Delta (1 \Gamma X n\Gamma1 )X n (n = 2; 3; \Delta \Delt... |

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- 1997
(Show Context)
Citation Context ...jV (B)) = V (B). However, as in the discussion around (16) and (17), Example 20 shows that it does not necessarily follow that P (N 2 BjV (C); C ` N) equals V (B), as it does in Examples 5 and 6. See =-=[36]-=- for some applications of this property in the setting of Example 6. It is natural to ask what additional hypothesis is appropriate for this stronger conclusion to hold in a more general setting, but ... |

3 |
Semi-stable Markov processes.I
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- 1972
(Show Context)
Citation Context ...subsets of R I . A real or vector-valued process (X t ; ts0) is called fi-self-similar where fi 2 R I if for every c ? 0 (X ct ; ts0) d = (c fi X t ; ts0) (20) Such processes were studied by Lamperti =-=[24, 25]-=-, who called them semistable. See [40] for a survey of the literature of these processes. A random closed subset Z of R I is self-similar in the sense (6) iff its age process is 1-selfsimilar. A natur... |

3 |
Ensembles régénératifs de la droite
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- 1983
(Show Context)
Citation Context ...ach non-degenerate F 0 . Among these is the following: Example 13 stationary-regen 0 (F 0 ). This is the unique distribution of a stationary 0 set Z that is regenerative subset of R I in the sense of =-=[26]-=-, and such that L has distribution F 0 . See Fristedt [13], who gives the following construction of Z, and further references. Let Z = f\GammaA 0 \Gamma Z 1 g [ fD 0 + Z 2 g (35) where A 0 and D 0 are... |

3 |
Some conditional expectation given an average of a stationary or self-similar random process
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(Show Context)
Citation Context ...g, but we do not have an answer to this question. In essence, the problem is the following: Problem 27 Find a condition that implies the identity (27) for a vectorvalued 0-self-similar process X. See =-=[37]-=- for a number of reformulations of (27) and further discussion, including a simple example of an R I 2 -valued 0-self-similar process X for which (27) fails to hold. We do not know much about rdd's de... |

3 |
Limit measures arising in the theory of groups
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Citation Context ...widely studied, motivated by a variety of applications including the following: ffl prior distributions in Bayesian statistics [12, 1], ffl asymptotic distributions in combinatorics and number theory =-=[42, 43, 41]-=-, ffl models for gene frequencies in population genetics [20, 6, 11] ffl models for species diversity in ecology [28, 10], ffl the representation of partition structures [21], ffl models for storage a... |

3 |
Limit measures arising in the theory of symmetric groups
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Citation Context ...widely studied, motivated by a variety of applications including the following: ffl prior distributions in Bayesian statistics [12, 1], ffl asymptotic distributions in combinatorics and number theory =-=[42, 43, 41]-=-, ffl models for gene frequencies in population genetics [20, 6, 11] ffl models for species diversity in ecology [28, 10], ffl the representation of partition structures [21], ffl models for storage a... |

2 |
Semilinear Markov processes, subordinators and renewal theory
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(Show Context)
Citation Context ...tionary strong-Markov process X. One can always take X to be the stationary version of the age process derived from the subordinator with L'evy measuresdescribed above. This is the method of Horowitz =-=[15]-=-. But zero sets of other Markov processes X may be considered. For example, the zero set Z of a stationary diffusion process X on the line, for which 0 is recurrent, gives a stationary-regen 0 set wit... |

1 |
The local time intensity of an exchangeable interval partition
- Kallenberg
- 1983
(Show Context)
Citation Context ...Z. One natural construction of such a Z, corresponding to an arbitrary prescribed distribution for (V n ), is obtained from the exchangeable interval partition considered by Berbee [3] and Kallenberg =-=[17]-=-. Here we consider constructions with a different sort of symmetry: Definition 3 self-sim 0 set. Call Z self-similar if Z d = cZ for all c ? 0; (6) where cZ = fcz; z 2 Zg, and d = denotes equality in ... |