| Title:
|
Discrete random processes with memory: Models and applications (English) |
| Author:
|
Kouřim, Tomáš |
| Author:
|
Volf, Petr |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
65 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
271-286 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
random walk |
| Keyword:
|
history dependent transition probability |
| Keyword:
|
non-Markov process |
| Keyword:
|
success punishing walk |
| Keyword:
|
success rewarding walk |
| MSC:
|
60G50 |
| MSC:
|
62F10 |
| idZBL:
|
07217110 |
| idMR:
|
MR4114252 |
| DOI:
|
10.21136/AM.2020.0335-19 |
| . |
| Date available:
|
2020-06-10T13:10:19Z |
| Last updated:
|
2022-07-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148143 |
| . |
| Reference:
|
[1] Davis, R. A., Liu, H.: Theory and inference for a class of nonlinear models with application to time series of counts.Stat. Sin. 26 (2016), 1673-1707. Zbl 1356.62137, MR 3586234, 10.5705/ss.2014.145t |
| Reference:
|
[2] Feller, W.: An Introduction to Probability Theory and Its Applications.A Wiley Publication in Mathematical Statistics. John Wiley & Sons, New York (1957). Zbl 0077.12201, MR 0088081 |
| Reference:
|
[3] Hawkes, A. G.: Spectra of some self-exciting and mutually exciting point processes.Biometrika 58 (1971), 83-90. Zbl 0219.60029, MR 0278410, 10.1093/biomet/58.1.83 |
| Reference:
|
[4] Kouřim, T.: Random walks with varying transition probabilities.Doktorandské dny 2017 P. Ambrož, Z. Masáková ČVUT, FJFI, Praha (2017), 141-150. |
| Reference:
|
[5] Kouřim, T.: Random walks with memory applied to grand slam tennis matches modeling.MathSport International 2019: Conference Proceedings Propobos Publications, Athens (2019), 220-227. |
| Reference:
|
[6] Kouřim, T.: Statistical Analysis, Modeling and Applications of Random Processes with Memory: PhD Thesis Study.ČVUT FJFI, Praha (2019). MR 4114252 |
| Reference:
|
[7] Pearson, K.: The problem of the random walk.Nature 72 (1905), 342 \99999JFM99999 36.0303.02. 10.1038/072342a0 |
| Reference:
|
[8] Rossi, R. J.: Mathematical Statistics: An Introduction to Likelihood Based Inference.John Wiley & Sons, Hoboken (2018). Zbl 1407.62006, 10.1002/9781118771075 |
| Reference:
|
[9] Schütz, G. M., Trimper, S.: Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk.Phys. Rev. E 70 (2004), Article ID 045101. 10.1103/PhysRevE.70.045101 |
| Reference:
|
[10] Turban, L.: On a random walk with memory and its relation with Markovian processes.J. Phys. A, Math. Theor. 43 (2010), Article ID 285006, 9 pages. Zbl 1204.82022, MR 2658904, 10.1088/1751-8113/43/28/285006 |
| . |
