scholarly journals A*-mixing convergence theorem for convex set valued processes

1987 ◽  
Vol 10 (1) ◽  
pp. 9-16
Author(s):  
A. de Korvin ◽  
R. Kleyle
Keyword(s):  
Banach Space ◽  
Convergence Theorem ◽  
Information Science ◽  
Convex Sets ◽  
Convex Set ◽  
Strong Law ◽  
Large Numbers ◽  
Mixing Sequences ◽  
Feedback Data ◽  
Bounded Convex

In this paper the concept of a*-mixing process is extended to multivalued maps from a probability space into closed, bounded convex sets of a Banach space. The main result, which requires that the Banach space be separable and reflexive, is a convergence theorem for*-mixing sequences which is analogous to the strong law of large numbers. The impetus for studying this problem is provided by a model from information science involving the utilization of feedback data by a decision maker who is uncertain of his goals. The main result is somewhat similar to a theorem for real valued processes and is of interest in its own right.

On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate

2013 ◽  
Vol 56 (2) ◽  
pp. 272-282 ◽  
Author(s):  
Lixin Cheng ◽  
Zhenghua Luo ◽  
Yu Zhou
Keyword(s):  
Banach Space ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Lower Semicontinuous ◽  
Weakly Compact ◽  
Bounded Set ◽  
Fréchet Differentiable ◽  
Bounded Convex

AbstractIn this note, we first give a characterization of super weakly compact convex sets of a Banach space X: a closed bounded convex set K ⊂ X is super weakly compact if and only if there exists a w* lower semicontinuous seminorm p with p ≥ σK ≌ supxєK 〈.,x〉 such that p2 is uniformly Fréchet differentiable on each bounded set of X*. Then we present a representation theoremfor the dual of the semigroup swcc(X) consisting of all the nonempty super weakly compact convex sets of the space X.

scholarly journals On Chung-Teicher type strong law for arrays of vector-valued random variables

2004 ◽  
Vol 2004 (9) ◽  
pp. 443-458
Author(s):  
Anna Kuczmaszewska
Keyword(s):  
Banach Space ◽  
Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Random Elements ◽  
Vector Valued

We study the equivalence between the weak and strong laws of large numbers for arrays of row-wise independent random elements with values in a Banach spaceℬ. The conditions under which this equivalence holds are of the Chung or Chung-Teicher types. These conditions are expressed in terms of convergence of specific series ando(1)requirements on specific weighted row-wise sums. Moreover, there are not any conditions assumed on the geometry of the underlying Banach space.

scholarly journals Mazur's intersection property of balls for compact convex sets

1987 ◽  
Vol 35 (2) ◽  
pp. 267-274 ◽  
Author(s):  
J. H. M. Whitfield ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Intersection Property ◽  
Compact Sets ◽  
Compact Convex Set ◽  
Mazur's Intersection Property

We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.

SOME LIMITING BEHAVIOR FOR ASYMPTOTICALLY NEGATIVE ASSOCIATED RANDOM VARIABLES

2016 ◽  
Vol 32 (1) ◽  
pp. 58-66 ◽  
Author(s):  
Qunying Wu ◽  
Yuanying Jiang
Keyword(s):  
Convergence Theorem ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Independent Random Variables ◽  
Strong Law ◽  
Limiting Behavior ◽  
Associated Random Variables ◽  
Large Numbers

In this paper, we study the almost sure convergence for sequences of asymptotically negative associated (ANA) random variables. As a result, we extend the classical Khintchine–Kolmogorov convergence theorem, Marcinkiewicz strong law of large numbers, and the three series theorem for sequences of independent random variables to sequences of ANA random variables without necessarily adding any extra conditions.

scholarly journals A Strong Law of Large Numbers for Random Sets in Fuzzy Banach Space

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
R. Ghasemi ◽  
A. Nezakati ◽  
M. R. Rabiei
Keyword(s):  
Metric Space ◽  
Convex Sets ◽  
Law Of Large Numbers ◽  
Compact Convex ◽  
Random Sets ◽  
Fuzzy Metric Space ◽  
Strong Law ◽  
Fuzzy Metric ◽  
Large Numbers ◽  
Fuzzy Random

The main purpose of this paper is to consider the strong law of large numbers for random sets in fuzzy metric space. Since many years ago, limited theorems have been expressed and proved for fuzzy random variables, but despite the uncertainty in fuzzy discussions, the nonfuzzy metric space has been used. Given that the fuzzy random variable is defined on the basis of random sets, in this paper, we generalize the strong law of large numbers for random sets in the fuzzy metric space. The embedded theorem for compact convex sets in the fuzzy normed space is the most important tool to prove this generalization. Also, as a result and by application, we use the strong law of large numbers for random sets in the fuzzy metric space for the bootstrap mean.

scholarly journals Strong law of large numbers forρ*-mixing sequences with different distributions

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Guang-Hui Cai
Keyword(s):  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Large Numbers ◽  
Mixing Sequences

Strong law of large numbers and complete convergence forρ*-mixing sequences with different distributions are investigated. The results obtained improve the relevant results by Utev and Peligrad (2003).

scholarly journals Laws of large numbers for infinite means with mixing sequences

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 783-793
Author(s):  
Jian Han ◽  
Xiaoqin Li ◽  
Yudan Cheng
Keyword(s):  
Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Mixing Sequences

In this paper, we consider the laws of large numbers with infinite means based on ?-mixing sequences. An exact weak law and a strong law are obtained for ?-mixing asymmetrical Cauchy random variables. It is also presented that the weak law cannot extend to a strong law. In addition, some simulations are presented to illustrate our results of the laws of large numbers.

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