Order statistics for jumps of normalised subordinators

Perman, Mihael

doi:10.1016/0304-4149(93)90007-q

Cited by 50 publications

(65 citation statements)

References 13 publications

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“…We refer to Perman [10] and section 8.1 in Pitman and Yor [15] for more information about these conditional laws. We also mention that a different fragmentation process has been constructed by Pitman [13], again time-reversing a remarkable coalescent process, cf.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A fragmentation process connected to Brownian motion

Bertoin¹

2000

Probab Theory Relat Fields

100

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Let (B s , s ≥ 0) be a standard Brownian motion and T 1 its first passage time at level 1. For every t ≥ 0, we consider ladder time set L (t) of the Brownian motion with drift t, B (t) s = B s + ts, and the decreasing sequence. .) of lengths of the intervals of the random partition of [0, T 1 ] induced by L (t) . The main result of this work is that (F (t), t ≥ 0) is a fragmentation process, in the sense that for 0 ≤ t < t , F (t ) is obtained from F (t) by breaking randomly into pieces each component of F (t) according to a law that only depends on the length of this component, and independently of the others. We identify the fragmentation law with the one that appears in the construction of the standard additive coalescent by Aldous and Pitman [3].

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Section: Introduction and Main Resultsmentioning

confidence: 99%

A fragmentation process connected to Brownian motion

Bertoin¹

2000

Probab Theory Relat Fields

100

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“…The well known fact the structural distribution of pd(θ) is beta(1, θ) follows immediately from (20). It follows easily from any one of the previous general formulas (27), (35), (36) or (40), that the EPPF of a pd(θ)-partition Π = (Π n ) is given by the formula…”

Section: The One-parameter Poisson-dirichlet Distributionmentioning

confidence: 92%

“…Following McCloskey [37], Kingman [29], Engen [15], Perman-Pitman-Yor [40,41,50], consider the ranked random discrete distribution (P i ) := (J i /T ) derived from an inhomogeneous Poisson point process of random lengths J 1 ≥ J 2 ≥ · · · ≥ 0 by normalizing these lengths by their sum T := ∞ i=1 J i . So it is assumed that the number N I of J i that fall in an interval I is a Poisson variable with mean Λ(I), for some Lévy measure Λ on (0, ∞), and the counts N I 1 , · · · , N I k are independent for every finite collection of disjoint intervals I 1 , · · · , I k .…”

Section: The Poisson-kingman Modelmentioning

confidence: 99%

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Poisson-Kingman partitions

Pitman¹

2003

Institute of Mathematical Statistics Lecture Notes - Monograph Series

149

353

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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 two-parameter family of Poisson-Dirichlet 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.

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“…The limiting random variable V in Theorem 4.2 has a continuous distribution with support on (0, 1), see Perman [20]. This is also the distribution of the longest completed excursion of the standard Brownian motion during time interval [0, 1].…”

Section: The Longest Excursionmentioning

confidence: 99%

On randomly spaced observations and continuous-time random walks

Basrak

Špoljarić

2016

J. Appl. Probab.

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We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy tailed steps, the limiting behavior of extreme observations until a given time t tends to be rather involved. We describe this asymptotics and generalize several partial results which appeared in this setting. In contrast to the earlier studies, our analysis is based in the point processes theory. The theory is applied to determine the asymptotic distribution of maximal excursions and sojourn times for continuous time random walks.

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Order statistics for jumps of normalised subordinators

Cited by 50 publications

References 13 publications

A fragmentation process connected to Brownian motion

A fragmentation process connected to Brownian motion

Poisson-Kingman partitions

On randomly spaced observations and continuous-time random walks

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