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 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 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 Pólya Urns Via the Contraction Method

2014 ◽  
Vol 23 (6) ◽  
pp. 1148-1186 ◽  
Author(s):  
MARGARETE KNAPE ◽  
RALPH NEININGER
Keyword(s):  
Asymptotic Behaviour ◽  
Discrete Time ◽  
Normal Limit ◽  
Rooted Trees ◽  
Limit Laws ◽  
Contraction Method ◽  
Limit Distributions

We propose an approach to analysing the asymptotic behaviour of Pólya urns based on the contraction method. For this, a new combinatorial discrete-time embedding of the evolution of the urn into random rooted trees is developed. A decomposition of these trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each colour. Ideas from the contraction method are used to study such systems of recursive distributional equations asymptotically. We apply our approach to a couple of concrete Pólya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behaviour.

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.

Multivariate linear time series models

1984 ◽  
Vol 16 (3) ◽  
pp. 492-561 ◽  
Author(s):  
E. J. Hannan ◽  
L. Kavalieris
Keyword(s):  
Limit Theorems ◽  
Analysis Of Algorithms ◽  
Linear Time ◽  
Asymptotic Properties ◽  
Real Data ◽  
Rates Of Convergence ◽  
Structure Theory ◽  
Time Series Models ◽  
Transfer Model ◽  
Rational Transfer

This paper is in three parts. The first deals with the algebraic and topological structure of spaces of rational transfer function linear systems—ARMAX systems, as they have been called. This structure theory is dominated by the concept of a space of systems of order, or McMillan degree, n, because of the fact that this space, M(n), can be realised as a kind of high-dimensional algebraic surface of dimension n(2s + m) where s and m are the numbers of outputs and inputs. In principle, therefore, the fitting of a rational transfer model to data can be considered as the problem of determining n and then the appropriate element of M(n). However, the fact that M(n) appears to need a large number of coordinate neighbourhoods to cover it complicates the task. The problems associated with this program, as well as theory necessary for the analysis of algorithms to carry out aspects of the program, are also discussed in this first part of the paper, Sections 1 and 2.The second part, Sections 3 and 4, deals with algorithms to carry out the fitting of a model and exhibits these algorithms through simulations and the analysis of real data.The third part of the paper discusses the asymptotic properties of the algorithm. These properties depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record, T. The necessary limit theorems and the analysis of the algorithms are given in Section 5. Many of these results are of interest independent of the algorithms being studied.

scholarly journals Profil Darah Ikan Gelodok (Periophthalmodon schlosseri) dan (Boleophthalamus boddarti) di Desa Kuala Tambangan Pelaihari, Kalimantan Selatan

2020 ◽  
Vol 7 (2) ◽  
pp. 176
Author(s):  
Nita Azhari ◽  
Hidayaturrahmah Hidayaturrahmah
Keyword(s):  
Normal Limit ◽  
Hemoglobin Concentration ◽  
Mangrove Ecosystem ◽  
Normal Range ◽  
Blood Profile ◽  
Normal Limits ◽  
Range Type ◽  
Periophthalmodon Schlosseri ◽  
The Village ◽  
Mean Cell Hemoglobin

Profil darah memiliki peran yang sangat penting dalam fisiologi metabolisme dan aktifitas tubuh hewan. Kuala Tambangan memiliki banyak potensi sumber daya ikan salah satunya ikan gelodok. Ikan gelodok di kawasan ini mudah ditemukan, akan tetapi sampai sekarang belum dimanfaatkan dengan baik oleh masyarakat. Tujuan penelitian ini untuk mengetahui profil darah ikan gelodok jenis Periophthalmodon schlosseri dan Boleophthalamus boddarti pada ekosistem mangrove yang berada di desa Kuala Tambangan, Kabupaten Tanah Laut, Kalimantan Selatan. Metode yang digunakan pada penelitian ini yaitu metode penangkapan hewan langka yaitu line transek, metode hemositometer dan metode sahli parameter yang dihitung hemoglobin, eritrosit, leukosit, hematokrit, MCV (Mean Corpusculla Volume), MCH (Mean Cell Hemoglobin), MCHC (Mean Cell Hemoglobin Concentration) pada 2 jenis ikan gelodok 34 ekor P. schlosseri dan 34 ekor B. boddarti. Hasil yang didapatkan adalah eritrosit P. schlosseri 3,87±0,58 x 106 sel/μL; B. boddarti 3,78±0,73 x 106 sel/μL 2 jenis ikan gelodok ini memiliki nilai eritrosit diatas batas normal; leukosit P. schlosseri 11,91±5,61 x103 sel/μL dan B. boddartii 9,72±4,24 x103 sel/μL nilai leukosit pada 2 jenis ikan gelodok ini berada di atas batas normal; hemoglobin P. schlosseri 11,59±1,75 % dan B. boddartii 11,75±1,96 % dari hasil yang didapat kadar hemoglobin pada 2 jenis ikan gelodok ini berada di atas batas normal kadar hemoglobin ikan pada umumnya; hematokrit P. schlosseri 34,32±5,57 % dan B. boddartii 35,71±5,44 % hasil hematokrit yang didapat dari 2 jenis ikan gelodok ini yaitu di atas batas normal; jenis P. schlosseri memiliki nilai MCV 88,72±6,62 μm3 yang berada di bawah batas normal; MCH 29,92±0,69 pg/sel yang berada di bawah batas normal; MCHC 33,99±3,97 g/dL pada jenis ini nilai MCHC masih berada pada batas normal; jenis B. boddartii memiliki nilai MCV 96,16±17,96 μm3 yang berada di bawah batas normal; MCH 31,51±5,50 pg/sel yang masih berada di batas normal; MCHC 32,87±1,77 g/dL nilai MCHC pada jenis ini masih berada pada batas normal. Blood profile has a very important role in the physiology of metabolism and animal body activities. Kuala Tambangan has a lot of potential fish resources, one of which is the fish Mudskipper. Mudskipper fish in this area are easy to find, but until now it has not been utilized properly by the community. The purpose of this study was to determine the blood profile of the Periophthalmodon Schlosseri and Boleophthalamus boddarti fish species in the mangrove ecosystem in the village of Kuala Tambangan, Tanah Laut District, South Kalimantan . The method used in this study is the method of catching endangered animals namely trasnek line, hemocytometer method and parameter Sahli method which is calculated hemoglobin, erythrocytes, leukocytes, hematocrit, MCV (Mean Corpuscular Volume), MCH (Mean Cell Hemoglobin), MCHC (Mean Cell Hemoglobin Concentration) on 2 types of fish, 34 Periophthalmodon schlosseri and 34 Boleophthalmus boddarti. The results obtained were P. schlosseri erythrocytes 3.87 ± 0.58 x 106 cells / μL; B. boddarti 3.78 ± 0.73 x 106 cells / μL 2 types of fish Mudskipper has erythrocyte values above the normal limit; P. schlosseri leukocytes 11.91 ± 5.61 x103 cells / μL and B. boddartii 9.72 ± 4.24 x103 cells / μL leukocyte values in these 2 types of jagged fish are above normal limits; hemoglobin P. schlosseri 11.59 ± 1.75 % and B. boddartii 11.75 ± 1.96 % from the results obtained by the hemoglobin levels in these 2 types of fishes are above the normal limit of fish hemoglobin levels in general; hematocrit P.schlosseri 34.32 ± 5.57 % and B. boddartii 35.71 ± 5.44 % hematocrit results obtained from these 2 types of mudskipper fish are above normal limits; P.schlosseri species had MCV values of 88.72 ± 6.62 μm3 which were below normal limits; MCH 29.92 ± 0.69 pg / cell that is below the normal range; MCHC 33.99 ± 3.97 g / dL in this type the MCHC value is still in the normal range; type B. boddartii has a MCV value of 96.16 ± 17.96 μm3 which is below normal limits; MCH 31,51 ± 5.50 pg / cell which is still in the normal range; MCHC 32.87 ± 1.77 g / dL MCHC values in this type are still within normal limits.Keywords: gelodok, blood, kuala tambangan

Multivariate linear time series models

1984 ◽  
Vol 16 (03) ◽  
pp. 492-561 ◽  
Author(s):  
E. J. Hannan ◽  
L. Kavalieris
Keyword(s):  
Limit Theorems ◽  
Analysis Of Algorithms ◽  
Linear Time ◽  
Asymptotic Properties ◽  
Real Data ◽  
Rates Of Convergence ◽  
Structure Theory ◽  
Time Series Models ◽  
Transfer Model ◽  
Rational Transfer

This paper is in three parts. The first deals with the algebraic and topological structure of spaces of rational transfer function linear systems—ARMAX systems, as they have been called. This structure theory is dominated by the concept of a space of systems of order, or McMillan degree,n,because of the fact that this space,M(n), can be realised as a kind of high-dimensional algebraic surface of dimensionn(2s+m) wheresandmare the numbers of outputs and inputs. In principle, therefore, the fitting of a rational transfer model to data can be considered as the problem of determiningnand then the appropriate element ofM(n). However, the fact thatM(n) appears to need a large number of coordinate neighbourhoods to cover it complicates the task. The problems associated with this program, as well as theory necessary for the analysis of algorithms to carry out aspects of the program, are also discussed in this first part of the paper, Sections 1 and 2.The second part, Sections 3 and 4, deals with algorithms to carry out the fitting of a model and exhibits these algorithms through simulations and the analysis of real data.The third part of the paper discusses the asymptotic properties of the algorithm. These properties depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record,T. The necessary limit theorems and the analysis of the algorithms are given in Section 5. Many of these results are of interest independent of the algorithms being studied.

scholarly journals Dependence between path-length and size in random digital trees

2017 ◽  
Vol 54 (4) ◽  
pp. 1125-1143
Author(s):  
Michael Fuchs ◽  
Hsien-Kuei Hwang
Keyword(s):  
Path Length ◽  
Normal Limit ◽  
Limit Laws ◽  
Normal Distributions ◽  
Digital Trees ◽  
Bivariate Normal ◽  
Asymmetric Case ◽  
Periodic Fluctuations ◽  
Type Of Behavior ◽  
Small Periodic

AbstractWe study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries, that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose components have correlation either zero or one or periodically oscillating. Moreover, the same type of behavior is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.

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