Applications of Mathematics, Vol. 65, No. 3, pp. 271-286, 2020
Discrete random processes with memory: Models and applications
Tomáš Kouřim, Petr Volf
Received November 30, 2019. Published online May 25, 2020.
Abstract: The contribution focuses on Bernoulli-like random walks, where the past events significantly affect the walk's future development. The main concern of the paper is therefore the formulation of models describing the dependence of transition probabilities on the process history. Such an impact can be incorporated explicitly and transition probabilities modulated using a few parameters reflecting the current state of the walk as well as the information about the past path. The behavior of proposed random walks, as well as the task of their parameter estimation, are studied both theoretically and with the aid of simulations.
Keywords: random walk; history dependent transition probability; non-Markov process; success punishing walk; success rewarding walk
References: [1] R. A. Davis, H. Liu: Theory and inference for a class of nonlinear models with application to time series of counts. Stat. Sin. 26 (2016), 1673-1707. DOI 10.5705/ss.2014.145t | MR 3586234 | Zbl 1356.62137
[2] W. Feller: An Introduction to Probability Theory and Its Applications. A Wiley Publication in Mathematical Statistics. John Wiley & Sons, New York (1957). MR 0088081 | Zbl 0077.12201
[3] A. G. Hawkes: Spectra of some self-exciting and mutually exciting point processes. Biometrika 58 (1971), 83-90. DOI 10.1093/biomet/58.1.83 | MR 0278410 | Zbl 0219.60029
[4] T. Kouřim: Random walks with varying transition probabilities. Doktorandské dny 2017 (P. Ambrož, Z. Masáková, eds.). ČVUT, FJFI, Praha, 2017, 141-150.
[5] T. Kouřim: Random walks with memory applied to grand slam tennis matches modeling. MathSport International 2019: Conference Proceedings. Propobos Publications, Athens, 2019, 220-227.
[6] T. Kouřim: Statistical Analysis, Modeling and Applications of Random Processes with Memory: PhD Thesis Study. ČVUT FJFI, Praha (2019).
[7] K. Pearson: The problem of the random walk. Nature 72 (1905), 342. DOI 10.1038/072342a0 | JFM 36.0303.02
[8] R. J. Rossi: Mathematical Statistics: An Introduction to Likelihood Based Inference. John Wiley & Sons, Hoboken (2018). DOI 10.1002/9781118771075 | Zbl 1407.62006
[9] G. M. Schütz, S. Trimper: Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk. Phys. Rev. E 70 (2004), Article ID 045101. DOI 10.1103/PhysRevE.70.045101
[10] L. Turban: On a random walk with memory and its relation with Markovian processes. J. Phys. A, Math. Theor. 43 (2010), Article ID 285006, 9 pages. DOI 10.1088/1751-8113/43/28/285006 | MR 2658904 | Zbl 1204.82022
Affiliations: Tomáš Kouřim, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Praha 1, Czech Republic, e-mail: kourim@outlook.com; Petr Volf, Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 4, 182 00 Praha 8, Czech Republic, e-mail: volf@utia.cas.cz
