Multidimensional Interval Trees

Javanian, Mehri; Vahidi-Asl, Mohammad Q.

doi:10.1007/978-3-0348-7915-6_25

Cited by 1 publication

(1 citation statement)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The random trees in these three examples have also been studied by [15], [18] and [19], where the properties of a randomly selected branch are investigated. This problem is quite different, and there is no phase transition.…”

Section: Examplesmentioning

confidence: 99%

The size of random fragmentation trees

Janson

Neininger

2007

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

Abstract. We study a random fragmentation process and its associated random tree. The process has earlier been studied by Dean and Majumdar [7], who found a phase transition: the number of fragmentations is asymptotically normal in some cases but not in others, depending on the position of roots of a certain characteristic equation. This parallels the behaviour of discrete analogues with various random trees that have been studied in computer science. We give rigorous proofs of this phase transition, and add further details.The proof uses the contraction method. We extend some previous results for recursive sequences of random variables to families of random variables with a continuous parameter; we believe that this extension has independent interest. The problem and resultConsider the following fragmentation process [7]. Fix b ≥ 2 and a random vector V = (V 1 , . . . , V b ). Note that the definitions and results below depend only on the distribution of (V 1 , . . . , V b ), so it would be more precise to say that we fix a distribution on R b ; we find it, however, more convenient to state the results in terms of a random vector. We assume throughout the paper that 0 ≤ V j ≤ 1, j = 1, . . . , b, and( 1.1) i.e., that (V 1 , . . . , V b ) belongs to the standard simplex. For simplicity we also assume that each V j < 1 a.s. We allow V j = 0, but note that, a.s., 0 < V j < 1 for at least one j.Starting with an object of size x ≥ 1, we break it into b pieces with sizes V 1 x, . . . , V b x. Continue recursively with each piece of size ≥ 1, using new (independent) copies of the random vector (V 1 , . . . , V b ) each time. The process terminates a.s. after a finite number of steps, leaving a finite set of fragments of sizes < 1. We let N (x) be the random number of fragmentation events, i.e., the number of pieces of size ≥ 1 that appear during the process; further, let N e (x) be the final number of fragments, i.e., the number of pieces of size < 1 that appear.

show abstract

Section: Examplesmentioning

confidence: 99%