Numbers For Fuzzy Random Variables Research Articles

Strong Law of Large Numbers for Fuzzy Random Variables in Fuzzy Metric Space

Strong Law of Large Numbers for Fuzzy Random Variables in Fuzzy Metric Space

T-norm-based limit theorems for fuzzy random variables

The objective of this paper is to derive some limit theorems of fuzzy random variables under the extension principle associated with continuous Archimedean triangular norms (t-norms). First of all, some convergence theorems for the sum of fuzzy random variables in chance measure and expected value are proved respectively based on the arithmetics of continuous Archimedean triangular norms. Then, a law of large numbers for fuzzy random variables is established by using the obtained convergence theorems. The results of the derived law of large numbers can degenerate to the strong laws of large numbers for random variables and fuzzy variables, respectively.

On Chung's type law of large numbers for fuzzy random variables

Let { X n | n ⩾ 1 } be a sequence of independent fuzzy random variables and { a n | n ⩾ 1 } a sequence of positive real numbers converging to ∞ . In this paper we show that, under the assumption ∑ n = 1 ∞ E φ ( ∥ X n ∥ ρ p ) / φ ( a n ) < ∞ with some restrictions on φ , S n / a n → 0 ˜ a.s. if and only if S n / a n → 0 ˜ in probability if and only if S n / a n → 0 ˜ in L 1 where S n = ∑ i = 1 n X i . This will generalize earlier results of strong law of large numbers for random elements in a Banach space (Choi and Sung (1988). Bull. Austral. Math. Soc. 37, 93–100).

An embedding theorem for convex fuzzy sets

In this paper we embed the space of upper semicontinuous convex fuzzy sets on a Banach space into a space of continuous functions on a compact space. The following structures are preserved by the embedding: convex cone, metric, sup-semilattice. The indicator function of the unit ball is mapped to the constant function 1. Two applications are presented: strong laws of large numbers for fuzzy random variables and Korovkin type approximation theorems.

Weak laws of large numbers for fuzzy random variables

In this paper, we establish weak laws of large numbers for sums of convex-compactly uniformly integrable fuzzy random variables taking values in the space of fuzzy numbers in R p . As a result, weak laws of large numbers for sums of convexly tight fuzzy random variables are given.

An approach to generalize laws of large numbers for fuzzy random variables

The formulation of strong and weak laws of large numbers for fuzzy random variables based on variance is given. The concept of the variance of fuzzy random variables, introduced by Körner (Fuzzy Sets and Systems 92 (1997) 83) and by Feng (Fuzzy Sets and Systems 120 (2001) 487) demonstrates its efficiency.

On the convergence of fuzzy martingales

The purpose of this paper is to prove the convergence theorem for fuzzy martingales without assuming that their values are Lipschitz or continuous fuzzy numbers on R n . This approach allows us to deduce many results on convergence of fuzzy numbers and fuzzy random variables from the relevant results on real numbers and real-valued random variables that appear as their support functions. Based on the concept of fuzzy downward martingale and applications of convergence theorem of fuzzy martingales, the strong law of large numbers for fuzzy random variables is proved, here the convergence is in the uniform metric but not the separable metric on fuzzy number space.

Evaluate Fuzzy Riemann Integrals Using the Monte Carlo Method

Techniques for using the Monte Carlo method to evaluate fuzzy Riemann integrals and improper fuzzy Riemann integrals are proposed in this paper. Owing to the α-level set of the (improper) fuzzy Riemann integral being the closed interval whose end points are the classical (improper) Riemann integrals, it is possible to invoke the Monte Carlo method to approximate the end points of the α-level closed intervals. We develop the strong law of large numbers for fuzzy random variables in order to give the techniques proposed for evaluating the (improper) fuzzy Riemann integrals using the Monte Carlo approach more theoretical support. The membership function of the (improper) fuzzy Riemann integral can be transformed into mathematical programming problems. Therefore, we can obtain the membership value by solving the mathematical programming problems using the commercial optimizer.

Kolmogorov's strong law of large numbers for fuzzy random variables

In this paper, Kolmogorov's strong law of large numbers for sums of independent and level-wise identically distributed fuzzy random variables is obtained.

The laws of large numbers for fuzzy random variables

The new attempt of weak and strong law of large numbers for fuzzy random variables is discussed in this paper by proposing the convergence in probability and convergence with probability one for fuzzy random variables. We first consider the limit properties of fuzzy numbers by invoking the Hausdorff metric, and then we extend it to the convergence in probability and convergence with probability one for fuzzy random variables. We provide the notion of weak and strong convergence in probability and weak and strong convergence with probability one for fuzzy random variables. Finally we come up with the weak and strong law of large numbers for fuzzy random variables in weak and strong sense.

A theorem of renewal process for fuzzy random variables and its application

The fuzzy set was introduced by Zadeh (1965) and the concept of fuzzy random variables was provided by Kwakernaak (1981). Sequences of independent and identical distributed fuzzy random variables were considered by Kruse (1982). He also showed the strong law of large numbers for fuzzy random variables. In this paper, we consider the stochastic process for fuzzy random variables and prove a theorem for fuzzy rate of a fuzzy renewal process.

A strong law of large numbers for fuzzy random variables

In this paper, we obtain a strong law of large numbers for sums of levelwise independent and levelwise identically distributed fuzzy random variables.

Chung’s law of large numbers for fuzzy random variables

In this paper, we obtain a generalization of Chung’s strong law of large numbers to the case of independent fuzzy random variables.

Marcinkiewicz-type law of large numbers for fuzzy random variables

We generalize a theorem of Marcinkiewicz to fuzzy random variables.

A strong law of large numbers for fuzzy random variables

A limit of a sequence of fuzzy numbers is defined and its some properties are shown. Based on these concept and properties, an independent sequence of fuzzy random variables is considered and a strong law of large numbers for fuzzy random variables is shown.

The strong law of large numbers for fuzzy random variables

Sequences of independent and identically distributed fuzzy random variables are considered. It is shown that the strong law of large numbers holds also for fuzzy random variables. This result is used to give a consistent estimator for the expectation of a fuzzy random variable.