Verovkin H. Asymptotics of random processes with immigration

Dissertation for the scientific level of Doctor of Philosophy in specialty 124 — ”System analysis“. — Taras Shevchenko National University of Kyiv, Ministry of Education and Science of Ukraine, Kyiv, 2021. The thesis is devoted to the analysis of renewal shot noise processes and their generalizations called random processes with immigration at renewal epochs. Random processes with immigration constitute a wide class of random processes that describe various cumulative phenomena. The construction of random processes with immigration can be described as follows. Impulses arrive in a certain abstract system at times determined by successive jumps of a random walk on the positive half-line. Upon its arrival the impulse generates a response which is described by a certain stochastic process, called the response process. The value of a random process with immigration at a certain time is the total effect of all the impulses that have entered the system up to and including that time. In this model, the moments of successive jumps of a random walk are called renewal epochs, hence the name “random processes with immigration at renewal epochs”. In the case when the response process is a deterministic (non-random) function, we are dealing with renewal shot noise processes.