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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
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.
W. Feller: An Introduction to Probability Theory and Its Applications. A Wiley Publication in Mathematical Statistics. John Wiley & Sons, New York, 1957.
A. G. Hawkes: Spectra of some self-exciting and mutually exciting point processes. Biometrika 58 (1971), 83–90.
T. Kouřim: Random walks with varying transition probabilities. Doktorandské dny 2017 (P. Ambrož, Z. Masáková, eds.). ČVUT, FJFI, Praha, 2017, pp. 141–150.
T. Kouřim: Random walks with memory applied to grand slam tennis matches modeling. MathSport International 2019: Conference Proceedings. Propobos Publications, Athens, 2019, pp. 220–227.
T. Kouřim: Statistical Analysis, Modeling and Applications of Random Processes with Memory: PhD Thesis Study. ČVUT FJFI, Praha, 2019.
K. Pearson: The problem of the random walk. Nature 72 (1905), 342.
R. J. Rossi: Mathematical Statistics: An Introduction to Likelihood Based Inference. John Wiley & Sons, Hoboken, 2018.
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.
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.
The research was supported by the grant No. 18-02739S of the Grant Agency of the Czech Republic.
About this article
Cite this article
Kouřim, T., Volf, P. Discrete Random Processes with Memory: Models and Applications. Appl Math 65, 271–286 (2020). https://doi.org/10.21136/AM.2020.0335-19
- random walk
- history dependent transition probability
- non-Markov process
- success punishing walk
- success rewarding walk