Strong laws of large numbers for adapted arrays of set-valued and fuzzy-valued random variables in Banach space

Quang, Nguyễn Văn; Thuan, Nguyen Tran

doi:10.1016/j.fss.2012.04.012

Cited by 12 publications

(3 citation statements)

References 20 publications

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“…For the theory of set-valued random variables (that is, random variables that taking their values in the space of all non-empty closed subsets of a Banach space X), most results in probability theory in the case of single-valued random variables are expanded [24,22,6]. Research directions on set-valued random variables are often related to limit theorems, the law of large numbers, and martingales [32,25,15].…”

Section: Introductionmentioning

confidence: 99%

Weak Set-Valued Martingale Difference and Its Applications

Tuyen¹,

Vuong²,

Quynh³

et al. 2022

Int. J. of Appl. Math.

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The martingale difference is considered widely in finance and economic because of its application in efficient market, in which the conditional expectation E[d t |F t−1 ] = 0 a.s., ∀t ≥ 2 for a sequence of asset returns {d t , t ≥ 1} and related historical information F t−1 . However, the concept of martingale difference in set-valued random variables (i.e. random sets) has not been studied. This paper proves some properties of a set-valued random variable sequence called a weak set-valued martingale difference. By studying its characteristic properties, we propose a method of testing the weak set-valued martingale difference hypothesis and perform some simulations with real data.

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Section: Introductionmentioning

confidence: 99%

Weak Set-Valued Martingale Difference and Its Applications

Tuyen¹,

Vuong²,

Quynh³

et al. 2022

Int. J. of Appl. Math.

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“…A possible way to handle this type of 'data' is using the concepts of fuzzy set-valued random variable introduced by Puri and Ralescu [38]. Over past 40 years there were many important works for set-valued and fuzzy set-valued random variables related to strong law of large numbers [5,14,41], center limit theorems [1,26,27] and convergence of martingales [22,25] in plenty of areas such as in media, imaging, and data processing. Currently there is no proposal in the literature which works on set-valued (milti-valued) martingale difference sequences and tests the null hypothesis for that concept on real-life data.…”

Section: Introductionmentioning

confidence: 99%

On the Testing Multi-Valued Martingale Difference Hypothesis

Tuyen¹

2018

JCC

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This paper presents a definition of Multi-Valued Martingale Difference (MVMD) based on Castaing representation of a multi-valued martingale that consists of martingale difference selections. Testing the Multi-Valued Martingale Difference Hypothesis (MVMDH) then examined. Testing the Martingale Difference Hypothesis (MDH) earlier was based on linear measures then later developed two directions in order to account for the existing nonlinearity in economic and financial data. First, the classical approaches have been modified by take into account the possible nonlinear dependence. Second, the use of more sophisticated statistical tools such as those based generalized spectral analysis. According to this article, both these developments in MDH are modified for MVMDH and applies them to exchange rate data and returns of stock market data.

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“…• to classify fuzzy data (see, for instance, Coppi et al [1], Ferraro and Giordani [2] and Guillaume et al [3]), • to obtain some limit and probabilistic results for random fuzzy numbers (see, for instance, Colubi et al [4], Molchanov [5], Terán [6,7], Quang and Thuan [8], Aletti and Bongiorno [9]), • in optimization problems (see, for instance, Abbasbandy and Asady [10], Abbasbandy and Amirfakhrian [11], Prochelvi et al [12], Báez-Sánchez et al [13], Bana and Coroianu [14], Bera et al [17], Coroianu [15], Coroianu et al [16]) • and especially in performing many statistical analyses (see, for instance, Näther [18,19], Körner and Näther [20], Körner [21], García et al [22], Montenegro et al [23,24], Gil et al [25], Coppi et al [26], González-Rodríguez et al [29,30,31], Ferraro et al [27], Ferraro and Giordani [28], Ramos-Guajardo and Lubiano [32], Sinova et al [39]). In the literature on fuzzy numbers and more general fuzzy sets, several metrics have been suggested (see, for instance, Puri and Ralescu [33], Klement et al [34]).…”

Section: Introductionmentioning

confidence: 99%

A parameterized metric between fuzzy numbers and its parameter interpretation

Sinova

Alvarez

López

et al. 2014

Fuzzy Sets and Systems

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When handling fuzzy number data, it is common practice to make use of a metric to quantify distances between fuzzy numbers. Several metrics have been suggested in the literature for this purpose. When statistically analyzing fuzzy number-valued data, L 2 metrics become especially useful. This paper introduces a new family of generalized L 2 metrics which take into account key features of the involved fuzzy numbers, namely, a measure of central location and two measures associated with the shape of the fuzzy numbers are used. A crucial property related to these three measures is that necessary and sufficient conditions can be established for them to characterize fuzzy numbers. Furthermore, the family of generalized L 2 metrics depends on one parameter. A discussion is provided regarding the interpretation of this parameter which can guide selection of its value in practice. * corresponding author: magil@uniovi.es, Fax: (+34) 985103354. Preprint submitted to Fuzzy Sets and SystemsJune 7, 2013 AbstractWhen handling fuzzy number data, it is common practice to make use of a metric to quantify distances between fuzzy numbers. Several metrics have been suggested in the literature for this purpose. When statistically analyzing fuzzy number-valued data, L 2 metrics become especially useful. This paper introduces a new family of generalized L 2 metrics which take into account key features of the involved fuzzy numbers, namely, a measure of central location and two measures associated with the shape of the fuzzy numbers are used. A crucial property related to these three measures is that necessary and sufficient conditions can be established for them to characterize fuzzy numbers. Furthermore, the family of generalized L 2 metrics depends on one parameter. A discussion is provided regarding the interpretation of this parameter which can guide selection of its value in practice.Keywords: fuzzy number, L 2 -metric, wabl/ldev/rdev representation of fuzzy numbers, L 2 wabl/ldev/rdev metric, weighting parameter

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Strong laws of large numbers for adapted arrays of set-valued and fuzzy-valued random variables in Banach space

Cited by 12 publications

References 20 publications

Weak Set-Valued Martingale Difference and Its Applications

Weak Set-Valued Martingale Difference and Its Applications

On the Testing Multi-Valued Martingale Difference Hypothesis

A parameterized metric between fuzzy numbers and its parameter interpretation

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