scholarly journals The first descent in samples of geometric random variables and permutations

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Arnold Knopfmacher ◽  
Helmut Prodinger
Keyword(s):  
Random Variables ◽  
Random Permutation ◽  
International Audience

International audience For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first descent in the word. In addition we compute the average initial and end height of the first descent for a random permutation of n letters.

scholarly journals Random permutations and their discrepancy process

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Computational Biology ◽  
Symmetric Group ◽  
Gaussian Process ◽  
Brownian Bridge ◽  
Random Permutation ◽  
Partial Sums ◽  
Random Permutations ◽  
New Process ◽  
International Audience

International audience Let $\sigma$ be a random permutation chosen uniformly over the symmetric group $\mathfrak{S}_n$. We study a new "process-valued" statistic of $\sigma$, which appears in the domain of computational biology to construct tests of similarity between ordered lists of genes. More precisely, we consider the following "partial sums": $Y^{(n)}_{p,q} = \mathrm{card} \{1 \leq i \leq p : \sigma_i \leq q \}$ for $0 \leq p,q \leq n$. We show that a suitable normalization of $Y^{(n)}$ converges weakly to a bivariate tied down brownian bridge on $[0,1]^2$, i.e. a continuous centered gaussian process $X^{\infty}_{s,t}$ of covariance: $\mathbb{E}[X^{\infty}_{s,t}X^{\infty}_{s',t'}] = (min(s,s')-ss')(min(t,t')-tt')$.

scholarly journals The distribution of ascents of size $d$ or more in samples of geometric random variables

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Charlotte Brennan ◽  
Arnold Knopfmacher
Keyword(s):  
Nonnegative Integer ◽  
Random Variables ◽  
Limiting Distribution ◽  
Geometric Probability ◽  
International Audience ◽  
The Mean ◽  
Mean Variance

International audience We consider words or strings of characters $a_1a_2a_3 \ldots a_n$ of length $n$, where the letters $a_i \in \mathbb{Z}$ are independently generated with a geometric probability $\mathbb{P} \{ X=k \} = pq^{k-1}$ where $p+q=1$. Let $d$ be a fixed nonnegative integer. We say that we have an ascent of size $d$ or more if $a_{i+1} \geq a_i+d$. We determine the mean, variance and limiting distribution of the number of ascents of size $d$ or more in a random geometrically distributed word.

scholarly journals A phase transition in the random transposition random walk

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Nathanael Berestycki ◽  
Rick Durrett
Keyword(s):  
Phase Transition ◽  
Genome Rearrangement ◽  
Graph Model ◽  
Random Permutation ◽  
Surprising Result ◽  
Parsimony Method ◽  
Random Graph Model ◽  
Minimum Number ◽  
International Audience ◽  
Random Transposition

International audience Our work is motivated by Bourque-Pevzner's simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on $n$ elements starting from the identity. Let $D_t$ be the minimum number of transpositions needed to go back to the identity element from the location at time $t$. $D_t$ undergoes a phase transition: for $0 < c ≤ 1$, the distance $D_cn/2 ~ cn/2$, i.e., the distance increases linearly with time; for $c > 1$, $D_cn/2 ~ u(c)n$ where u is an explicit function satisfying $u(x) < x/2$. Moreover we describe the fluctuations of $D_{cn/2}$ about its mean at each of the three stages (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulation-fragmentation process and relating the behavior to the Erdős-Rényi random graph model.

scholarly journals A product formula for the TASEP on a ring

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Erik Aas ◽  
Jonas Sjöstrand
Keyword(s):  
Stationary Distribution ◽  
Random Permutation ◽  
Product Formula ◽  
Combinatorial Interpretation ◽  
International Audience

International audience For a random permutation sampled from the stationary distribution of the TASEP on a ring, we show that, conditioned on the event that the first entries are strictly larger than the last entries, the $\textit{order}$ of the first entries is independent of the $\textit{order}$ of the last entries. The proof uses multi-line queues as defined by Ferrari and Martin, and the theorem has an enumerative combinatorial interpretation in that setting. Finally, we present a conjecture for the case where the small and large entries are not separated. Pour une permutation randomisée tirée de la mesure stationnaire du TASEP, nous démontrons, conditionnée à l’évènement que les premières lettres sont plus grandes que les dernières lettres, que l’ordre des petites lettres est indépendant de l’ordre des grandes lettres. La preuve utilise les files d’attente multilignes de Ferrari et Martin, et le théorème a une interprétation combinatoire énumérative dans ce contexte. Finalement, nous présentons une conjecture pour le cas où les petits et les grandes lettres ne sont pas séparées.

scholarly journals Note on the weighted internal path length of b-ary trees

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Limit Theorem ◽  
Continuous Time ◽  
Path Length ◽  
Analysis Of Algorithms ◽  
Random Variables ◽  
Time Parameter ◽  
Recursive Sequences ◽  
International Audience

Analysis of Algorithms International audience In a recent paper Broutin and Devroye (2005) have studied the height of a class of edge-weighted random trees.This is a class of trees growing in continuous time which includes many wellknown trees as examples. In this paper we derive a limit theorem for the internal path length for this class of trees.For the proof we extend a limit theorem in Neininger and Rüschendorf (2004) to recursive sequences of random variables with continuous time parameter.

Approximations to extreme tail probabilities for sampling without replacement

1969 ◽  
Vol 66 (3) ◽  
pp. 587-606 ◽  
Author(s):  
M. Stone
Keyword(s):  
Random Sample ◽  
Normal Approximation ◽  
Random Variables ◽  
Tail Probability ◽  
Random Permutation ◽  
Simple Random Sample ◽  
Tail Probabilities ◽  
Asymptotically Normal ◽  
Sampling Fraction ◽  
Image Position

Introduction and summary: In the distribution of the sum of n random variablesfor which we suppose , the tail probability will be called an extreme tail probability if the normal approximationis poor, when is asymptotically normal and n is large. This paper concerns the specialization of (1.1) in whichwith a random permutation of nλ ones and n − nλ zeros, that is, in which T is the total for a simple random sample from the population with sampling fraction λ. In this case, we will always write T as Tλ. For brevity, we may write {c1,…, cn} for .

scholarly journals Limit laws for a class of diminishing urn models.

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Random Variables ◽  
Urn Models ◽  
Limit Laws ◽  
Sampling Without Replacement ◽  
International Audience ◽  
Special Instance ◽  
Special Case

International audience In this work we analyze a class of diminishing 2×2 Pólya-Eggenberger urn models with ball replacement matrix M given by $M= \binom{ -a \,0}{c -d}, a,d∈\mathbb{N}$ and $c∈\mathbb{N} _0$. We obtain limit laws for this class of 2×2 urns by giving estimates for the moments of the considered random variables. As a special instance we obtain limit laws for the pills problem, proposed by Knuth and McCarthy, which corresponds to the special case $a=c=d=1$. Furthermore, we also obtain limit laws for the well known sampling without replacement urn, $a=d=1$ and $c=0$, and corresponding generalizations, $a,d∈\mathbb{N}$ and $c=0$.

scholarly journals Common intervals in permutations

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Sylvie Corteel ◽  
Guy Louchard ◽  
Robin Pemantle
Keyword(s):  
Generating Function ◽  
Random Permutation ◽  
International Audience ◽  
Bounded Size

International audience An interval of a permutation is a consecutive substring consisting of consecutive symbols. For example, 4536 is an interval in the permutation 71453682. These arise in genetic applications. For the applications, it makes sense to generalize so as to allow gaps of bounded size δ -1, both in the locations and the symbols. For example, 4527 has gaps bounded by 1 (since 3 and 6 are missing) and is therefore a δ -interval of 389415627 for δ =2. After analyzing the distribution of the number of intervals of a uniform random permutation, we study the number of 2-intervals. This is exponentially large, but tightly clustered around its mean. Perhaps surprisingly, the quenched and annealed means are the same. Our analysis is via a multivariate generating function enumerating pairs of potential 2-intervals by size and intersection size.\par

scholarly journals Volume Laws for Boxed Plane Partitions and Area Laws for Ferrers Diagrams

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Uwe Schwerdtfeger
Keyword(s):  
Uniform Distribution ◽  
Random Variables ◽  
Symmetry Class ◽  
Limit Laws ◽  
Plane Partitions ◽  
Symmetric Plane ◽  
International Audience ◽  
Ferrers Diagrams ◽  
Area Limit

International audience We asymptotically analyse the volume random variables of general, symmetric and cyclically symmetric plane partitions fitting inside a box. We consider the respective symmetry class equipped with the uniform distribution. We also prove area limit laws for two ensembles of Ferrers diagrams. Most limit laws are Gaussian.

scholarly journals Infinite Systems of Functional Equations and Gaussian Limiting Distributions

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Michael Drmota ◽  
Bernhard Gittenberger ◽  
Johannes F. Morgenbesser
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Combinatorial Enumeration ◽  
Limiting Distributions ◽  
Infinite Dimensional ◽  
Infinite Systems ◽  
Enumeration Problems ◽  
International Audience

International audience In this paper infinite systems of functional equations in finitely or infinitely many random variables arising in combinatorial enumeration problems are studied. We prove sufficient conditions under which the combinatorial random variables encoded in the generating function of the system tend to a finite or infinite dimensional limiting distribution.

Sign in / Sign up

Export Citation Format

Share Document