Abstract
The Wiener–Khinchin transformation is examined. It is noted that there are stochastic processes which do not simultaneously have a correlation function that is typical for a stationary process, and a power spectral density. We determine the dependence of the statistical stability on the power spectral density of the process and investigate the statistical stability of a process for which the power spectral density is described by a power function. Results are obtained for continuous and discrete processes. We then present simulation results which confirm the correctness of the formulas describing the dependence of the statistical instability parameters on the power spectral density of the process. The dependence of the statistical stability of a process on its correlation characteristics is analyzed. The statistical stability of low frequency and narrowband stochastic processes is investigated.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Gorban, I.I.: Statistically unstable processes: links with flicker, nonequilibrium, fractal, and color noise. Radioelectron. Commun. Syst. 55(3), 99–114 (2012)
Gorban, I.I.: Fenomen Statisticheskoy Ustoichivosti (The Phenomenon of Statistical Stability). Naukova dumka, Kiev (2014)
Gorban, I.I.: Statisticheskaya ustoychivost nizkochastotnykh i polosovykh shumov (Statistical stability for low-frequency and band noise). Math. Mach. Syst. 2, 104–112 (2015a)
Gorban, I.I.: Statisticheskaya ustoychivost sluchaynykh protsesov (Statistical stability of random processes). Math. Mach. Syst. 3, 100–111 (2015b)
Gorban, I.I.: Sluchaynost i gipersluchaynost (Randomness and Hyper-randomness). Naukova dumka, Kiev (2016) Â
Jahnke, E., Emde, F., Lösch, F.: Tafeln Höherer Funktionen. B.G. Teubner Verlagsgesellschaft, Stuttgart (1960)
Kharkevich, A.A.: Lineynye i Nelineynye sistemy (Linear and Non-linear systems). Nauka, Moscow (1973)
Prudnikov, A.P., Brychkov, Yu. A., Marichev, O.I.: Integraly i Ryady. Elementarnye Funktsii (Integrals and Series. Elementary Functions), vol. 1. Nauka, Moscow (1981)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Gorban, I.I. (2017). Dependence of the Statistical Stability of a Stochastic Process on Its Spectrum-Correlation Characteristics. In: The Statistical Stability Phenomenon. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-43585-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-43585-5_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-43584-8
Online ISBN: 978-3-319-43585-5
eBook Packages: EngineeringEngineering (R0)