scholarly journals A survey of multivariate aspects of the contraction method

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Ralph Neininger ◽  
Ludger Rüschendorf
Keyword(s):  
Limit Theorems ◽  
Analysis Of Algorithms ◽  
Branching Processes ◽  
Random Trees ◽  
Open Problems ◽  
Limit Laws ◽  
Contraction Method ◽  
Recursive Sequences ◽  
International Audience ◽  
General Conditions

International audience We survey multivariate limit theorems in the framework of the contraction method for recursive sequences as arising in the analysis of algorithms, random trees or branching processes. We compare and improve various general conditions under which limit laws can be obtained, state related open problems and give applications to the analysis of algorithms and branching recurrences.

scholarly journals Exponential bounds and tails for additive random recursive sequences

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Analysis Of Algorithms ◽  
Sufficient Conditions ◽  
Random Trees ◽  
Divide And Conquer ◽  
Recursive Sequences ◽  
Recursive Structures ◽  
International Audience ◽  
Exponential Bounds ◽  
The Moment

Analysis of Algorithms International audience Exponential bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We derive tail bounds from estimates of the Laplace transforms and of the moment sequences. For the proof we use some classical exponential bounds and some variants of the induction method. The paper generalizes results of Rösler (% \citeyearNPRoesler:91, % \citeyearNPRoesler:92) and % \citeNNeininger:05 on subgaussian tails to more general classes of additive random recursive sequences. It also gives sufficient conditions for tail bounds of the form \exp(-a t^p) which are based on a characterization of \citeNKasahara:78.

scholarly journals On rates of convergence in the probabilistic analysis of algorithms

2021 ◽  
Author(s):  
◽  
Jasmin Straub
Keyword(s):  
Normal Limit ◽  
Analysis Of Algorithms ◽  
Rates Of Convergence ◽  
Random Trees ◽  
Divide And Conquer ◽  
Complexity Measures ◽  
Limit Laws ◽  
Contraction Method ◽  
Normal Limits ◽  
Recursive Structures

Within the last thirty years, the contraction method has become an important tool for the distributional analysis of random recursive structures. While it was mainly developed to show weak convergence, the contraction approach can additionally be used to obtain bounds on the rate of convergence in an appropriate metric. Based on ideas of the contraction method, we develop a general framework to bound rates of convergence for sequences of random variables as they mainly arise in the analysis of random trees and divide-and-conquer algorithms. The rates of convergence are bounded in the Zolotarev distances. In essence, we present three different versions of convergence theorems: a general version, an improved version for normal limit laws (providing significantly better bounds in some examples with normal limits) and a third version with a relaxed independence condition. Moreover, concrete applications are given which include parameters of random trees, quantities of stochastic geometry as well as complexity measures of recursive algorithms under either a random input or some randomization within the algorithm.

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.

scholarly journals Asymptotic Behaviour of Continuous Time and State Branching Processes

2000 ◽  
Vol 68 (1) ◽  
pp. 68-84 ◽  
Author(s):  
Zeng-Hu Li
Keyword(s):  
Asymptotic Behaviour ◽  
Continuous Time ◽  
Limit Theorems ◽  
Branching Processes ◽  
Limit Laws ◽  
Classical Setting

AbstractWe prove some limit theorems for contiunous time and state branching processes. The non-degenerate limit laws are obtained in critical and non-critical cases by conditioning or introducing immigration processes. The limit laws in non-critical cases are characterized in terms of the cononical measure of the cumulant semigroup. The proofs are based on estimates of the cumulant semigroup derived from the forward and backward equations, which are easier than the proffs in the classical setting.

scholarly journals Branching processes in random environment die slowly

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Vladimir Vatutin ◽  
Andreas Kyprianou
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Generating Functions ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Limit Distributions ◽  
Conditional Limit Theorems ◽  
International Audience ◽  
Conditional Limit

International audience Let $Z_n,n=0,1,\ldots,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_0(s),f_1(s),\ldots,$ and let $S_0=0$, $S_k=X_1+ \ldots +X_k,k \geq 1$, be the associated random walk with $X_i=\log f_{i-1}^{\prime}(1), \tau (m,n)$ be the left-most point of minimum of $\{S_k,k \geq 0 \}$ on the interval $[m,n]$, and $T=\min \{ k:Z_k=0\}$. Assuming that the associated random walk satisfies the Doney condition $P(S_n > 0) \to \rho \in (0,1), n \to \infty$, we prove (under the quenched approach) conditional limit theorems, as $n \to \infty$, for the distribution of $Z_{nt}, Z_{\tau (0,nt)}$, and $Z_{\tau (nt,n)}, t \in (0,1)$, given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt$.

scholarly journals Polynomial tails of additive-type recursions

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Eva-Maria Schopp
Keyword(s):  
Tail Probability ◽  
Upper Bounds ◽  
Random Trees ◽  
Divide And Conquer ◽  
Variation Theory ◽  
Recursive Sequences ◽  
Recursive Structures ◽  
International Audience ◽  
Asymptotic Tail Probability ◽  
Polynomial Bounds

International audience Polynomial bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees, or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We mainly focuss on polynomial tails that arise due to heavy tail bounds of the toll term and the starting distributions. Besides estimating the tail probability directly we use a modified version of a theorem from regular variation theory. This theorem states that upper bounds on the asymptotic tail probability can be derived from upper bounds of the Laplace―Stieltjes transforms near zero.

scholarly journals A survey of limit laws for bootstrapped sums

2003 ◽  
Vol 2003 (45) ◽  
pp. 2835-2861 ◽  
Author(s):  
Sándor Csörgő ◽  
Andrew Rosalsky
Keyword(s):  
Limit Theorems ◽  
Open Problems ◽  
Limit Laws ◽  
First Order ◽  
Efron’S Bootstrap ◽  
The Relationship ◽  
Efron's Bootstrap ◽  
Independent And Identically Distributed

Concentrating mainly on independent and identically distributed (i.i.d.) real-valued parent sequences, we give an overview of first-order limit theorems available for bootstrapped sample sums for Efron's bootstrap. As a light unifying theme, we expose by elementary means the relationship between corresponding conditional and unconditional bootstrap limit laws. Some open problems are also posed.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

scholarly journals Limit theorems for multi-type general branching processes with population dependence

2020 ◽  
Vol 52 (4) ◽  
pp. 1127-1163
Author(s):  
Jie Yen Fan ◽  
Kais Hamza ◽  
Peter Jagers ◽  
Fima C. Klebaner
Keyword(s):  
Carrying Capacity ◽  
Population Model ◽  
General Framework ◽  
Limit Theorems ◽  
Mating Systems ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Special Section ◽  
Large Numbers ◽  
Type Population

AbstractA general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population as a measure-valued process and obtain its asymptotics as the population grows with the environmental carrying capacity. Thus, a deterministic approximation is given, in the form of a law of large numbers, as well as a central limit theorem. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.

Sign in / Sign up

Export Citation Format

Share Document