arXiv subjects
Publications and source records attributed to A. A. Yambartsev.
We establish a large deviation principle for the trajectories of Wiener processes subject to random resets to the origin occurring according to a Poisson process. In addition to the pathwise large deviation principle, we identify the rate function and establish a large deviation principle for the supremum of the process over long time intervals.
Bruss's odds theorem \cite{Bruss1} addresses the problem of determining the optimal stopping time for sequences of independent indicator functions. In this note, we derive upper and lower bounds for the success probability under the optimal stopping rule. These bounds depend on the number of independent events under consideration and on a deterministic index specifying the stopping time. Moreover, the bounds are sharp: we provide explicit examples in which the corresponding inequalities are attained with equality.
Birth-death processes form a natural class where ideas and results on large deviations can be tested. In this paper, we derive a large deviation principle under the assumption that the rate of a jump down (death) is growing asymptotically linearly with the population size, while the rate of a jump up (birth) is growing sub-linearly. We establish a large deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large deviation probabilities. The results show interesting features of dependence of the large deviation functional upon the parameters of the process and the forms of scaling and normalization.
The symmetric birth and death process in the integers $\{1, \ldots, N \}$ with linear rates is studied. The process moves slowly and spends more time in the neighborhood of the state 1. It represents our attempt at explaining the asymmetry between amounts of matter and antimatter by the inhomogeneity of the sojourn time. The state of the process reflects a relative frequency of an antimatter amount. The observed matter-antimatter imbalance in the Universe we consider a result of stochastic competition between them. The amount of matter significantly exceeds the amount of antimatter, which corresponds to the lower states of the process, and the path toward matter-antimatter equilibrium can be very long.
This work is a continuation of [7]. We consider a continuous-time birth-and-death process in which the transition rates have an asymptotical power-law dependence upon the position of the process. We establish rough exponential asymptotic for the probability that a sample path of a normalized process lies in a neighborhood of a given nonnegative continuous function. We propose a variety of normalization schemes for which the large deviation functional preserves its natural integral form.
The paper considers a continuous-time birth-death process where the jump rate has an asymptotically polynomial dependence on the process position. We obtain a rough exponential asymptotics for the probability of excursions of a re-scaled process contained within a neighborhood of a given continuous non-negative function.