Celaleddin Şençimen

Strong Statistical Convergence in Probabilistic Metric Spaces

Abstract In this article, we introduce the concepts of strongly statistically convergent sequence and strong statistically Cauchy sequence in a probabilistic metric (PM) space endowed with the strong topology, and establish some basic facts. Next, we define the strong statistical limit points and the strong statistical cluster points of a sequence in this space and investigate the relations between these concepts.

On weak ideal convergence in normed spaces

Abstract Recently, we have introduced the concepts of weak ideal convergence (briefly, weak I–convergence) and weak I–Cauchy sequence in a normed space, and investigated their basic properties [25]. In this study, we introduce the concepts of weak I*–convergence, weak I*–Cauchy sequence in a normed space, and present some main results.

Statistical continuity in probabilistic normed spaces

In this study, we investigate the statistical continuity in a probabilistic normed space. In this context, the statistical continuity properties of the probabilistic norm, the vector addition and the scalar multiplication are examined.

Statistically D-bounded sequences in probabilistic normed spaces

In this study, the concept of a statistically D-bounded sequence in a probabilistic normed (PN) space endowed with the strong topology is introduced and its basic properties are investigated. It is shown that a strongly statistically convergent sequence and a strong statistically Cauchy sequence are statistically D-bounded under certain conditions. A sequence which goes far away from the limit point infinitely many times and presents random deviations in a PN space may be handled with the tools of strong statistical convergence and statistical D-boundedness.

On exhaustive families of random functions and certain types of convergence

A random function of a variable t in T (or, a random function on T) is known as a function f whose values are random variables all defined on a common probability space, where T is an arbitrary set. A random function is also called a stochastic (random) process. In this work, we base ourselves on a random function of E-process type and such a function is also called a random function, briefly. In this approach, the domain of such a random function is or an interval of , and the set of values of this random function is considered as a special probabilistic metric (PM) space (more precisely, an E-space) of metric space-valued random variables, and all our definitions and results are presented using the tools of PM spaces. In this context, we introduce the concept of an exhaustive family of such random functions, which is a natural generalization of equicontinuity, and we investigate its basic properties. We also examine some of the properties related to the continuous convergence in probability for a sequence of such random functions and certain conditions which give rise to the continuity in probability of the limit of a sequence of such random functions.

Random metric space of fuzzy numbers

In this paper, we introduce the concept of a random metric (RM) space of fuzzy numbers and establish some of its basic properties. In this approach, the distance between two fuzzy numbers is considered as a certain measurable function. Using this concept, we also obtain some convergence relations for a sequence of fuzzy numbers. Finally, we investigate a boundedness property for a subset of such an RM space.