Asymptotics of random evolution on a renewal process

Abstract

This article is devoted to the asymptotic analysis of random evolutions constructed on the basis of a regenerating process. The study focuses on a class of stochastic processes with nonlinear time normalization, which arise in models that involve structural regeneration or memory effects. These processes are important for describing complex systems whose behavior restarts or changes at random time points. 
Particular attention is paid to the derivation of a renewal-type equation for the expecta-tion of the random evolution defined on the regenerating process under a nonlinear transfor-mation of time. The obtained equation characterizes the dynamics of the mean values of the evolution in terms of the regeneration function that reflects the probabilistic structure of the underlying process.
A limit theorem is proved, which establishes the asymptotic behavior of the expected values of the random evolution as time tends to infinity. It is shown that the nonlinear normal-ization significantly affects the nature of the asymptotic regime, influencing both the growth rate and the form of the asymptotic representation. Special attention is given to the case where the time normalization is defined by a monotonic increasing function, allowing for modeling acceleration or deceleration effects in the evolution.
The theoretical results obtained in this work can be applied to the analysis of models in reliability theory, queueing systems, population dynamics, and various applied areas where randomness, regeneration, and nonlinearity in time evolution play a crucial role. The devel-oped approach provides a general framework for studying the long-term behavior of systems driven by random inputs and regenerating mechanisms. 
Additionally, the results lay the groundwork for further generalizations to multidimen-sional random evolutions and more complex forms of time scaling in stochastic systems.