On exhaustive families of random functions and certain types of convergence

Abstract

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.

Keywords::

• random function
• continuous convergence in probability
• equicontinuity in probability
• exhaustiveness in probability
• Alexandroff convergence in probability

Keywords::

• 60G05
• 54E70

Acknowledgements

The authors are grateful to the referees whose valuable comments greatly improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

^1. Email: [email protected]