LAWS OF LARGE NUMBERS AND CONSISTENCY OF STATISTICAL ESTIMATORS IN ARBITRARY SPACES

Abstract

The laws of large numbers consist in that the empirical means converge to the theoretical ones. In the classical case, the arithmetic sample means under some conditions converge in probability, as the number of addends grow, to the mathematical expectation. The essence of the laws of large numbers is that they usually establish the consistency of a large body of statistical estimators. By and large, this topic seems to hold the lead in probability theory and mathematical statistics. Nevertheless, behind the mathematical apparatus used are properties of sums of random variables (or, generally speaking, elements of a linear space). Therefore, it cannot be applied to probabilistic problems related to objects of an arbitrary nature. These are binary relations, fuzzy sets, and, in general, the elements of spaces not endowed with a vector-like structure; they emerge rather frequently in applied studies (see [1–3]). In this connection, it seems pertinent to establish laws of large numbers in spaces of a nonnumerical nature. The following problems thus have to be solved. (A) To define the empirical mean. (B) To define the theoretical mean. (C) To introduce the convergence of the empirical means to the theoretical one. (D) To prove, under some conditions, the convergence of the empirical means to the theoretical one. (E) To justify the consistency of various statistical estimators. (F) To apply the results obtained to actual problems. The authors have been studying this topic since the 70s (see [4, 5]). But only recently have we succeeded in establishing the law of large numbers under quite natural constraints. This constitutes the dominant bulk of the present paper. In addition to generalizing [6], we give the results of computer analysis of the set of empirical means.