Large Numbers For Random Sets Research Articles

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

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.

Some theoretical results of learning theory based on random sets in set‐valued probability space

PurposeThe purpose of this paper is to introduce some basic knowledge of statistical learning theory (SLT) based on random set samples in set‐valued probability space for the first time and generalize the key theorem and bounds on the rate of uniform convergence of learning theory in Vapnik, to the key theorem and bounds on the rate of uniform convergence for random sets in set‐valued probability space. SLT based on random samples formed in probability space is considered, at present, as one of the fundamental theories about small samples statistical learning. It has become a novel and important field of machine learning, along with other concepts and architectures such as neural networks. However, the theory hardly handles statistical learning problems for samples that involve random set samples.Design/methodology/approachBeing motivated by some applications, in this paper a SLT is developed based on random set samples. First, a certain law of large numbers for random sets is proved. Second, the definitions of the distribution function and the expectation of random sets are introduced, and the concepts of the expected risk functional and the empirical risk functional are discussed. A notion of the strict consistency of the principle of empirical risk minimization is presented.FindingsThe paper formulates and proves the key theorem and presents the bounds on the rate of uniform convergence of learning theory based on random sets in set‐valued probability space, which become cornerstones of the theoretical fundamentals of the SLT for random set samples.Originality/valueThe paper provides a studied analysis of some theoretical results of learning theory.

On a uniform law of large numbers for random sets and subdifferentials of random functions

In this paper, we strengthen the convergence in a uniform law of large numbers for random upper semicontinuous multifunctions of Shapiro and Xu. The proof is based on an abstract law of large numbers in a metric space endowed with a convex combination operation. Convergence in the Hausdorff metric is obtained, whereas the original result presented a weakened form of convergence of excess functionals. As a consequence, another law of large numbers for subdifferentials of random functions is improved as well.

A law of large numbers for random sets

We show a law of large numbers for random sets taking values in a class of subsets of a Banach space, which is at least larger than the class of compact subsets of the space, if the Banach space is of infinite dimension. To see this we first define the Hausdorff distance on the family of all the non-empty subsets of a Banach space and give some properties of it. Motivated by the fact that a study of fuzzy sets is to study of set-valued analysis.

A class of bounds for convex bodies in Hilbert space

We extend to infinite dimensions a class of bounds forLp metrics of finite-dimensional convex bodies. A generalization to arbitrary increasing convex functions is done simultaneously. The main tool is the use of Gaussian measure to effect a normalization for varying dimension. At a point in the proof we also invoke a strong law of large numbers for random sets to produce a rotational averaging.

Strong Law of Large Numbers for Banach Space Valued Random Sets

In this paper we prove a strong law of large numbers for random sets whose values are compact convex subsets of a Banach space.

Law of Large Numbers for Random Sets and Allocation Processes

In this paper we establish a strong law of large numbers for unbounded random sets, and then apply it to an optimization problem arising in allocation processes under uncertainty.

A Strong Law of Large Numbers for Random Compact Sets

Abstract : In the study of probabilities on geometrical objects, there have been some recent attempts to formulate general theories of random sets, notably by Kendall and Matheron. It is the purpose here to make a contribution in this direction by demonstrating the existence of a strong law of large numbers for random sets taking values in the class of compact subsets of (R sup n). The result is proved first under the assumption of convexity and then extended to the general case.