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
Classification MSC:  60G50, 62F10

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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:; 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:

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