Convergence Almost Everywhere of Orthorecursive Expansions in Systems of Translates and Dilates

2018 ◽  
pp. 3-11
Author(s):  
Vladimir V. Galatenko ◽  
Taras P. Lukashenko ◽  
Victor A. Sadovnichiy
Keyword(s):  
Convergence Almost Everywhere ◽  
Almost Everywhere

scholarly journals Strong Law of Large Numbers of the Offspring Empirical Measure for Markov Chains Indexed by Homogeneous Tree

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Huilin Huang
Keyword(s):  
Markov Chains ◽  
Law Of Large Numbers ◽  
Empirical Measure ◽  
Homogeneous Tree ◽  
Almost Everywhere Convergence ◽  
Strong Law ◽  
Convergence Almost Everywhere ◽  
Almost Everywhere ◽  
Large Numbers

We study the limit law of the offspring empirical measure and for Markov chains indexed by homogeneous tree with almost everywhere convergence. Then we prove a Shannon-McMillan theorem with the convergence almost everywhere.

A Martingale Convergence Theorem in Vector Lattices

1966 ◽  
Vol 18 ◽  
pp. 424-432 ◽  
Author(s):  
Ralph DeMarr
Keyword(s):  
Convergence Theorem ◽  
Vector Lattice ◽  
Random Variables ◽  
Order Convergence ◽  
Arbitrary Vector ◽  
Martingale Convergence Theorem ◽  
Convergence Almost Everywhere ◽  
Vector Lattices ◽  
Almost Everywhere ◽  
Martingale Convergence

The martingale convergence theorem was first proved by Doob (3) who considered a sequence of real-valued random variables. Since various collections of real-valued random variables can be regarded as vector lattices, it seems of interest to prove the martingale convergence theorem in an arbitrary vector lattice. In doing so we use the concept of order convergence that is related to convergence almost everywhere, the type of convergence used in Doob's theorem.

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