Asymptotic distribution theory for Hoare's selection algorithm

1996 ◽  
Vol 28 (1) ◽  
pp. 252-269 ◽  
Author(s):  
Rudolf Grübel ◽  
Uwe Rösler
Keyword(s):  
Stochastic Process ◽  
Random Walk ◽  
Asymptotic Behaviour ◽  
Asymptotic Distribution ◽  
Random Environment ◽  
Binary Tree ◽  
Selection Algorithm ◽  
Real Numbers ◽  
Asymptotic Distribution Theory ◽  
Path Lengths

We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds the lth smallest in a set of n numbers. Letting n tend to infinity and considering the values l = 1, ···,n simultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree.

Asymptotic distribution theory for Hoare's selection algorithm

1996 ◽  
Vol 28 (01) ◽  
pp. 252-269 ◽  
Author(s):  
Rudolf Grübel ◽  
Uwe Rösler
Keyword(s):  
Stochastic Process ◽  
Random Walk ◽  
Asymptotic Behaviour ◽  
Asymptotic Distribution ◽  
Random Environment ◽  
Binary Tree ◽  
Selection Algorithm ◽  
Real Numbers ◽  
Asymptotic Distribution Theory ◽  
Path Lengths

We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds thelth smallest in a set ofnnumbers. Lettingntend to infinity and considering the valuesl =1, ···,nsimultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree.

scholarly journals Bindweeds or random walks in random environments on multiplexed trees and their asympotics

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Mikhail Menshikov ◽  
Dimitri Petritis ◽  
Serguei Popov
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Random Environment ◽  
Degrees Of Freedom ◽  
Random Variables ◽  
Internal Degree ◽  
Degree Of Freedom ◽  
Random Environments ◽  
Vertex Set ◽  
Internal Degree Of Freedom

International audience We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree.The term multiplexed means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal degree of freedom from the finite set $\{1,...,d\}$, for some integer $d$. The consequence of the internal degree of freedom is an enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexed by the set $\{1,...,d\} × \{1,...,d\}.$ This indexing conveys the information on the internal degree of freedom of the vertices contiguous to each edge. The term random environment means that the jumping rates for the random walk are a family of edge-indexed random variables, independent of the natural filtration generated by the random variables entering in the definition of the random walk; their joint distribution depends on the index of each component of the multi-edges. We study the large time asymptotic behaviour of this random walk and classify it with respect to positive recurrence or transience in terms of a specific parameter of the probability distribution of the jump rates.This classifying parameter is shown to coincide with the critical value of a matrix-valued multiplicative cascade on the ordinary tree (i.e.the one without internal degrees of freedom attached to the vertices) having the same vertex set as the state space of the random walk. Only results are presented here since the detailed proofs will appear elsewhere.

scholarly journals Ballisticity conditions for random walk in random environment

2010 ◽  
Vol 150 (1-2) ◽  
pp. 61-75 ◽  
Author(s):  
A. Drewitz ◽  
A. F. Ramírez
Keyword(s):  
Random Walk ◽  
Random Environment

scholarly journals Random walk in a high density dynamic random environment

2014 ◽  
Vol 25 (4) ◽  
pp. 785-799 ◽  
Author(s):  
Frank den Hollander ◽  
Harry Kesten ◽  
Vladas Sidoravicius
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
High Density ◽  
Density Dynamic

scholarly journals A Double-indexed Functional Hill Process and Applications

2016 ◽  
Vol 8 (4) ◽  
pp. 144
Author(s):  
Modou Ngom ◽  
Gane Samb Lo
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Stochastic Processes ◽  
Order Statistics ◽  
Finite Dimension ◽  
Domain Of Attraction ◽  
Extreme Value ◽  
Extreme Value Index

<div>Let $X_{1,n} \leq .... \leq X_{n,n}$ be the order statistics associated with a sample $X_{1}, ...., X_{n}$ whose pertaining distribution function (\textit{df}) is $F$. We are concerned with the functional asymptotic behaviour of the sequence of stochastic processes</div><div> </div><div>\begin{equation}<br />T_{n}(f,s)=\sum_{j=1}^{j=k}f(j)\left( \log X_{n-j+1,n}-\log<br />X_{n-j,n}\right)^{s} ,  \label{fme}<br />\end{equation}</div><div> </div><div>indexed by some classes $\mathcal{F}$ of functions $f:\mathbb{N}%^{\ast}\longmapsto \mathbb{R}_{+}$ and $s \in ]0,+\infty[$ and where $k=k(n)$ satisfies</div><div> </div><div>\begin{equation*}<br />1\leq k\leq n,k/n\rightarrow 0\text{ as }n\rightarrow \infty .<br />\end{equation*}</div><div> </div><div>We show that this is a stochastic process whose margins generate estimators of the extreme value index when $F$ is in the extreme domain of attraction. We focus in this paper on its finite-dimension asymptotic law and provide a class of new estimators of the extreme value index whose performances are compared to analogous ones. The results are next particularized for one explicit class $\mathcal{F}$.</div>

scholarly journals Random Walk Based Key Nodes Discovery in Opportunistic Networks

2016 ◽  
Vol 12 (03) ◽  
pp. 28
Author(s):  
Qin Qin ◽  
Yong-qiang He
Keyword(s):  
Random Walk ◽  
Binary Tree ◽  
Opportunistic Networks ◽  
Communication Cost ◽  
Simulation Experiments ◽  
Efficiency And Effectiveness ◽  
Iteration Number ◽  
Key Nodes

In opportunistic networks, temporary nodes choose neighbor nodes to forward messages while communicating. However, traditional forward mechanisms don’t take the importance of nodes into consideration while forwarding. In this paper, we assume that each node has a status indicating its importance, and temporary nodes choose the most important neighbors to forward messages. While discovering important neighbors, we propose a binary tree random walk based algorithm. We analyze the iteration number and communication cost of the proposed algorithm, and they are much less than related works. The simulation experiments validate the efficiency and effectiveness of the proposed algorithm.

scholarly journals Random walk in cooling random environment: ergodic limits and concentration inequalities

2019 ◽  
Vol 24 (0) ◽  
Author(s):  
Luca Avena ◽  
Yuki Chino ◽  
Conrado da Costa ◽  
Frank den Hollander
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Concentration Inequalities
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