Limit Theorems For Sequences Of Fuzzy Random Variables

When dealing with data, uncertainty is mainly incorporated in the sense of randomness. But it is reasonable to suppose that data also inherit uncertainty in the sense of vagueness, due to subjective judgement, imprecise human knowledge and perception. In order to integrate vagueness into techniques of statistical inference, recent developments during the last two decades propose to represent data by fuzzy subsets rather than real numbers or tuples of real numbers. Extending the basic notion of random variables, the properties of fuzzy random variables has been investigated.The extension of mathematical statistics to vague data may be regarded as reasonable and well founded if analogues of classical limit theorems like the strong law of large numbers and the convergence theorems of series are formulated. The classical limit theorems base, simultaneously, on a fixed notion of random variable according to a fixed metric space R. Some convergence properties are obtained by the specific characteristics of this metric space. The aim of this paper is to give analogues of the classical limit theorems which related to some different concepts of convergence and based on different metric spaces. In the uniform metric D, related subject has been widely discussed. Seldom results appear, however, with respect to sendograph and D₂^* metric. This paper mainly deals with the latter two metrics.In sendograph metric, a criterion of almost sure convergence for fuzzy random variables in sendograph metric is established. Ogura and Li have obtained another criterion which leads to a similar result. They show that the sendograph convergence follows from the convergence of the sequences of the level sets for rational-number levels, which are random sets, nevertheless will cause some limitations. Our criterion is an improved version, which turns that condition into the pointwise convergence of the real-valued random variables. As an application, one dominated convergence theorem in standard probability theory is extended to the case of fuzzy random variables. A part of Marcinkiewicz-Zygmund's type strong law of large numbers - and Kolmogorov's type, as a trivial corollary - is proved in the field of fuzzy random theory using the same technique.One paramount concern in standard probability theory is the behavior of series, i.e. sums of independent random variables. Kolmogorov three series theorem is of particularly importance. This paper deals with an ana- logue of it with respect to D₂^* metric. A variance series of independent random variables is included in the classical theorem. Under our circumstance, the definition of variance is based on the separable metric D₂^*, hence D₂^* is more reasonable than the metric D in the discussion of 2nd-order moment of fuzzy random variables in reference to Feng. In this paper, we first obtain a separable and complete metric space (E|^₂^d, D₂^*) by modifying the definition of fuzzy numbers using Yan's method. Next, the relationships among different types of convergence like mean-square convergence, almost sure convergence and convergence in probability are investigated. Our version of Kolmogorov three series theorem is finally proved by applying the above relationships and the properties of (E_^2^d, D₂^*).