On central limit and iterated logarithm supplements to the martingale convergence theorem

1977 ◽  
Vol 14 (4) ◽  
pp. 758-775 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Central Limit ◽  
Convergence Theorem ◽  
Mean Square ◽  
Martingale Convergence Theorem ◽  
Iterated Logarithm ◽  
Pólya Urn ◽  
Integrable Martingale ◽  
Polya Urn ◽  
Urn Scheme ◽  
Martingale Convergence

Let {Sn, n ≧ 1} be a zero, mean square integrable martingale for which so that Sn → S∞ a.s., say, by the martingale convergence theorem. The paper is principally concerned with obtaining central limit and iterated logarithm results for Bn(Sn – S∞) where the multipliers Bn ↑ ∞ a.s. An example on the Pólya urn scheme is given to illustrate the results.

On central limit and iterated logarithm supplements to the martingale convergence theorem

1977 ◽  
Vol 14 (04) ◽  
pp. 758-775 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Central Limit ◽  
Convergence Theorem ◽  
Mean Square ◽  
Martingale Convergence Theorem ◽  
Iterated Logarithm ◽  
Pólya Urn ◽  
Integrable Martingale ◽  
Polya Urn ◽  
Urn Scheme ◽  
Martingale Convergence

Let {Sn , n ≧ 1} be a zero, mean square integrable martingale for which so that Sn → S ∞ a.s., say, by the martingale convergence theorem. The paper is principally concerned with obtaining central limit and iterated logarithm results for Bn (Sn – S∞ ) where the multipliers Bn ↑ ∞ a.s. An example on the Pólya urn scheme is given to illustrate the results.

scholarly journals Central limit theorems for generalized Pólya urn models

2006 ◽  
Vol 43 (4) ◽  
pp. 938-951 ◽  
Author(s):  
I. Higueras ◽  
J. Moler ◽  
F. Plo ◽  
M. San Miguel
Keyword(s):  
Covariance Matrix ◽  
Central Limit ◽  
Limit Theorems ◽  
Lyapunov Equation ◽  
Central Limit Theorems ◽  
Random Trees ◽  
Urn Models ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn

In this paper we obtain central limit theorems for generalized Pólya urn models with L ≥ 2 colors where one out of K different replacements (actions) is applied randomly at each step. Each possible action constitutes a row of the replacement matrix, which can be nonsquare and random. The actions are chosen following a probability distribution given by an arbitrary function of the proportions of the balls of different colors present in the urn. Moreover, under the same hypotheses it is proved that the covariance matrix of the asymptotic distribution is the solution of a Lyapunov equation, and a procedure is given to obtain the covariance matrix in an explicit form. Some applications of these results to random trees and adaptive designs in clinical trials are also presented.

scholarly journals Beta-Stacy processes and a generalization of the Pólya-urn scheme

1997 ◽  
Vol 25 (4) ◽  
pp. 1762-1780 ◽  
Author(s):  
Stephen Walker ◽  
Pietro Muliere
Keyword(s):  
Pólya Urn ◽  
Polya Urn ◽  
Urn Scheme

scholarly journals Martingale Convergence Theorem for the Conditional Intuitionistic Fuzzy Probability

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1707
Author(s):  
Katarína Čunderlíková
Keyword(s):  
Fuzzy Sets ◽  
Conditional Probability ◽  
Convergence Theorem ◽  
Intuitionistic Fuzzy Sets ◽  
Fuzzy Probability ◽  
Martingale Convergence Theorem ◽  
Intuitionistic Fuzzy ◽  
Mv Algebra ◽  
Fuzzy State ◽  
Martingale Convergence

For the first time, the concept of conditional probability on intuitionistic fuzzy sets was introduced by K. Lendelová. She defined the conditional intuitionistic fuzzy probability using a separating intuitionistic fuzzy probability. Later in 2009, V. Valenčáková generalized this result and defined the conditional probability for the MV-algebra of inuitionistic fuzzy sets using the state and probability on this MV-algebra. She also proved the properties of conditional intuitionistic fuzzy probability on this MV-algebra. B. Riečan formulated the notion of conditional probability for intuitionistic fuzzy sets using an intuitionistic fuzzy state. We use this definition in our paper. Since the convergence theorems play an important role in classical theory of probability and statistics, we study the martingale convergence theorem for the conditional intuitionistic fuzzy probability. The aim of this contribution is to formulate a version of the martingale convergence theorem for a conditional intuitionistic fuzzy probability induced by an intuitionistic fuzzy state m. We work in the family of intuitionistic fuzzy sets introduced by K. T. Atanassov as an extension of fuzzy sets introduced by L. Zadeh. We proved the properties of the conditional intuitionistic fuzzy probability.

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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