Dependence

2021 ◽  
pp. 271-289
Author(s):  
James Davidson
Keyword(s):  
Ergodic Theorem ◽  
Fundamental Idea ◽  
Shift Transformation

A number of probabilistic approaches to the concept of dependence in stochastic sequences are contrasted. The fundamental idea is a shift transformation. Invariant and remote events and the idea of ergodicity are introduced and the ergodic theorem is proved. The final sections introduce the contrasting idea of dependence on sub-σ‎-fields with the notions of regularity and strong and uniform mixing defined among other variants.

Mean ergodic theorem in function symmetric spaces for infinite measure

2018 ◽  
Vol 2018 (1) ◽  
pp. 35-46
Author(s):  
Vladimir Chilin ◽  
◽  
Aleksandr Veksler ◽  
Keyword(s):  
Ergodic Theorem ◽  
Symmetric Spaces ◽  
Infinite Measure ◽  
Mean Ergodic Theorem

Probability Theory for Fuzzy Quantum Spaces with Statistical Applications

2017 ◽  
Author(s):  
Renáta Bartková ◽  
Beloslav Riečan ◽  
Anna Tirpáková
Keyword(s):  
Probability Theory ◽  
Fuzzy Sets ◽  
Ergodic Theorem ◽  
Intuitionistic Fuzzy Sets ◽  
Mathematics Students ◽  
Intuitionistic Fuzzy ◽  
Individual Ergodic Theorem ◽  
Recurrence Theorem ◽  
The Individual ◽  
Atanassov Intuitionistic Fuzzy Sets

The reference considers probability theory in two main domains: fuzzy set theory, and quantum models. Readers will learn about the Kolmogorov probability theory and its implications in these two areas. Other topics covered include intuitionistic fuzzy sets (IF-set) limit theorems, individual ergodic theorem and relevant statistical applications (examples from correlation theory and factor analysis in Atanassov intuitionistic fuzzy sets systems, the individual ergodic theorem and the Poincaré recurrence theorem). This book is a useful resource for mathematics students and researchers seeking information about fuzzy sets in quantum spaces.

The Fundamental Idea of State Socialism.Kimio Hayaski

1938 ◽  
Vol 43 (4) ◽  
pp. 687-688
Author(s):  
Hymen Ezra Cohen
Keyword(s):  
Fundamental Idea

Feasibility of Longest Prefix Matching using Learned Index Structures

2021 ◽  
Vol 48 (4) ◽  
pp. 45-48
Author(s):  
Shunsuke Higuchi ◽  
Junji Takemasa ◽  
Yuki Koizumi ◽  
Atsushi Tagami ◽  
Toru Hasegawa
Keyword(s):  
Machine Learning ◽  
Data Structure ◽  
Index Structures ◽  
Packet Forwarding ◽  
Fundamental Idea ◽  
Search Operation ◽  
Longest Prefix Matching ◽  
Ip Packet ◽  
Computation Speed

This paper revisits longest prefix matching in IP packet forwarding because an emerging data structure, learned index, is recently presented. A learned index uses machine learning to associate key-value pairs in a key-value store. The fundamental idea to apply a learned index to an FIB is to simplify the complex longest prefix matching operation to a nearest address search operation. The size of the proposed FIB is less than half of an existing trie-based FIB while it achieves the computation speed nearly equal to the trie-based FIB. Moreover, the computation speed of the proposal is independent of the length of IP prefixes, unlike trie-based FIBs.

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

2021 ◽  
pp. 1-36
Author(s):  
ARIE LEVIT ◽  
ALEXANDER LUBOTZKY
Keyword(s):  
Ergodic Theorem ◽  
Abelian Groups ◽  
Wreath Products ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Amenable Groups ◽  
Pointwise Ergodic Theorem ◽  
Lamplighter Group ◽  
Invariant Random Subgroups

Abstract We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

Non-singular -actions: an ergodic theorem over rectangles with application to the critical dimensions

2020 ◽  
pp. 1-18
Author(s):  
ANTHONY H. DOOLEY ◽  
KIERAN JARRETT
Keyword(s):  
Ergodic Theorem ◽  
Natural Action ◽  
Critical Dimensions ◽  
Measure Spaces ◽  
Metric Isomorphism

Abstract We adapt techniques developed by Hochman to prove a non-singular ergodic theorem for $\mathbb {Z}^d$ -actions where the sums are over rectangles with side lengths increasing at arbitrary rates, and in particular are not necessarily balls of a norm. This result is applied to show that the critical dimensions with respect to sequences of such rectangles are invariants of metric isomorphism. These invariants are calculated for the natural action of $\mathbb {Z}^d$ on a product of d measure spaces.

Induced transformations of recurrent a.m.s. dynamical systems

2015 ◽  
Vol 15 (02) ◽  
pp. 1550010
Author(s):  
Sheng Huang ◽  
Mikael Skoglund
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Random Process ◽  
Ergodic Theorem ◽  
Finite Measure ◽  
Stationary Random Process ◽  
Finite State ◽  
Ratio Ergodic Theorem

This note proves that an induced transformation with respect to a finite measure set of a recurrent asymptotically mean stationary dynamical system with a sigma-finite measure is asymptotically mean stationary. Consequently, the Shannon–McMillan–Breiman theorem, as well as the Shannon–McMillan theorem, holds for all reduced processes of any finite-state recurrent asymptotically mean stationary random process. As a by-product, a ratio ergodic theorem for asymptotically mean stationary dynamical systems is presented.

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