A model of discrete random walk with history-dependent transition probabilities

Abstract

This contribution deals with a model of one-dimensional Bernoulli-like random walk with the position of the walker controlled by varying transition probabilities. These probabilities depend explicitly on the previous move of the walker and, therefore, implicitly on the entire walk history. Hence, the walk is not Markov. The article follows on the recent work of the authors, the models presented here describe how the logits of transition probabilities are changing in dependence on the last walk step. In the basic model this development is controlled by parameters. In the more general setting these parameters are allowed to be time-dependent. The contribution focuses mainly on reliable estimation of model components via the MLE procedures in the framework of the generalized linear models.