We establish consistent estimators of jump positions and jump altitudes of a multi-level step function that is the best -approximation of a probability density function . If itself is a step-function the number of jumps may be unknown.
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...