Casual Stability of Some Systems of Random Variables

TL;DR

This paper explores the concept of casual stability in additive and multiplicative systems of random variables, extending the definition to systems with random sizes and operations like min and max, advancing the understanding of self-similar stochastic systems.

Contribution

It introduces modified definitions of casual stability for various systems of random variables, including multiplicative, min/max operations, and systems with random element counts.

Findings

Extended casual stability to multiplicative systems

Applied casual stability to systems with min/max operations

Analyzed systems with random number of elements

Abstract

Self-similarity of systems is very popular and intensively developing field during last decades. To this field belong so-called stable distributions and their generalization. In Klebanov and Sl\'amov\'a (2014) there was given an approach to define additive systems with the property of random self-similarity - casual stability (c.s.). Here we continue study the notion of casual stability for additive systems of random variables (r.v.). We also give a modification of this definition and spread them on multiplicative systems of r.v. and on the system with operations of taking minimum or maximum of r.v. The case of systems with a random number of elements is also considered.