Convergence in Distribution for Uncertain Random Sequences with Dependent Random Variables
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Convergence in Distribution for Uncertain Random Sequences with Dependent Random Variables∗ GAO Rong · AHMADZADE Hamed
DOI: 10.1007/s11424-020-9192-y Received: 28 June 2019 / Revised: 4 January 2020 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2020 Abstract Random variables and uncertain variables are respectively used to model randomness and uncertainty. While randomness and uncertainty always coexist in a same complex system. As an evolution of random variables and uncertain variables, uncertain random variable is introduced as a tool to deal with complex phenomena including randomness and uncertainty simultaneously. For uncertain random variables, a basic and important topic is to discuss the convergence of its sequence. Specifically, this paper focuses on studying the convergence in distribution for a sequence of uncertain random sequences with different chance distributions where random variables are not independent. And the result of this paper is a generalization of the existing literature. Relations among convergence theorems are studied. Furthermore, the theorems are explained by several examples. Keywords variable.
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Chance distribution, convergence in distribution, convergence in mean, uncertain random
Introduction
With the development of probability theory, we usually use random variables to tackle with noises around us under the assumption that these random variables are independent. While, in many practical stochastic models, the assumption of independence among the random variables is not plausible. In fact, increases in some random variables are usually related to decreases in other random variables, so the assumption of negative dependence is sometimes more appropriate than the assumption of independence. In the real life, different types of indeterminacy always coexist in a same system, for example, randomness and fuzziness usually occurred simultaneously. For model this hybrid phenomenon, Puri and Ralescu[1] presented the GAO Rong (Corresponding author) School of Economics and Management, Hebei University of Technology, Tianjin 300401, China. Email: [email protected]. AHMADZADE Hamed Department of Mathematical Sciences, University of Sistan and Baluchestan, Zahedan, Iran. ∗ This research was supported by the Natural Science Foundation of Hebei Province under Grant No. F2020202056, Key Project of Hebei Education Department under Grant No. ZD2020125. This paper was recommended for publication by Editor JIA Yingmin.
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GAO RONG · AHMADZADE HAMED
concept of fuzzy random variable and some of its properties were studied such as expected value and variance. Furthermore, in order to model dependent hybrid phenomena, Ahmadzade, et al.[2] introduced the concept of dependent fuzzy random variable and proved several convergence theorems for such fuzzy random variables. To our best knowledge, probability theory are used to describe randomness relative to frequencies. When we use probability, we always make the fundamental assumption that the estimated probability and the long-run cumula
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