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Integrated random pulse process with positive and negative periodicity
A V Kargovsky1, O A Chichigina1
1Faculty of Physics and International Laser Center, Lomonosov Moscow State University, Leninskie Gory, 119991 Moscow, Russia.
Physical Review. E
|September 16, 2022
Summary
This study analyzes nonstationary processes, finding that integrated renewal processes combine linear time functions with Wiener processes. Intensity depends on pulse process waiting times, impacting super-Poisson and sub-Poisson behaviors.
Area of Science:
- Stochastic Processes
- Probability Theory
Background:
- Nonstationary processes are crucial in various scientific fields.
- Understanding integrated stationary random sequences is key to modeling complex systems.
Purpose of the Study:
- To analyze nonstationary processes derived from stationary random sequences of delta pulses.
- To represent integrated renewal processes and their intensity.
- To investigate the behavior of cumulants and bounded variance processes.
Main Methods:
- Mathematical analysis of integrated renewal processes.
- Derivation of intensity based on mean and variance of waiting times.
- Proof of linear growth for cumulants over time.
Main Results:
- Integrated renewal processes are sums of linear time functions and Wiener processes.
- Process intensity is higher for super-Poisson than sub-Poisson processes.
- All cumulants exhibit linear growth over time; bounded variance for fixed intervals.
Conclusions:
- The study provides a clear representation of integrated renewal processes.
- Analytical findings align well with numerical simulations.
- The research offers insights into the behavior of nonstationary random processes.
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