ASYMPTOTICS OF A MIXTURE OF MULTIDIMENSIONAL RENEWAL EQUATIONS UNDER NONLINEAR NORMALIZATION

Abstract

This paper is devoted to the study of multidimensional renewal equations that arise in the analysis of stochastic processes and have applications in mathematical statistics, reliability theory, economics, and complex physical systems. The focus is on the case where these equations are considered within a nonlinear approximation framework. The main idea involves time normalization by means of an infinitesimal nonlinear factor depending on a small parameter tending to zero. Such a rescaling significantly extends the domain of investigation and allows for a more accurate description of the process dynamics, particularly in detecting short-term changes, jumps, and state transitions.

The renewal equations are formulated in matrix form, which makes it possible to incorporate the internal structure and interrelations among the components of the vector-valued process. The study considers processes governed by Markovian dynamics as well as those with independent increments. Within this framework, a model is constructed as a mixture of two multidimensional renewal equations. For each of these components, all necessary conditions for solvability and consistency are provided, the probabilities of their occurrence are specified, and the corresponding renewal functions are defined.

Particular attention is given to the representation of the solutions to these equations in terms of conditional expectations, which reveals the connection between the asymptotic behavior of the process and the structure of the renewal equations. As a result, for the proposed mixture model, a limiting representation is obtained under the assumption of weak convergence as the small parameter tends to zero. This leads to the formulation and proof of a limit theorem that describes the asymptotic behavior of the mixture of multidimensional renewal equations with nonlinear normalizing factors.