Affiliations: Department of Mathematics, University of California,
Los Angeles, CA 90095-1555, USA | TOBB Economics and Technology University, Faculty of
Arts and Sciences, Department of Mathematics, Söǧütözü
06530, Ankara, Turkey
Abstract: Statistical convergence was introduced in connection with problems
of series summation. Only later it was demonstrated that statistical
convergence is closely related to convergence of the main statistical
characteristics. Statistical limits are defined relaxing conditions on
conventional convergence. The main idea of the statistical convergence of a
sequence l is that the majority of elements from
l converge and we do not care what is going on with other
elements. At the same time, it is known that sequences that come from real life
sources, such as measurement and computation, do not allow, in a general case,
to test whether they converge or statistically converge in the strict
mathematical sense. To overcome these limitations, fuzzy convergence was
introduced earlier in the context of neoclassical analysis and fuzzy
statistical convergence is introduced and studied in this paper. We find
relations between fuzzy statistical convergence of a sequence and fuzzy
statistical convergence of its subsequences (Theorem 2.1), as well as between
fuzzy statistical convergence of a sequence and conventional convergence of its
subsequences (Theorem 2.2). It is demonstrated what operations with fuzzy
statistical limits are induced by operations on sequences (Theorem 2.3) and how
fuzzy statistical limits of different sequences influence one another (Theorem
2.4). In Section 3, relations between fuzzy statistical convergence and fuzzy
convergence of statistical characteristics, such as the mean (average) and
standard deviation, are studied (Theorems 3.1 and 3.2).
Keywords: Statistical convergence, fuzzy sets, fuzzy limits, statistics, mean, standard deviation, fuzzy convergence, fuzzy density
