Modern Stochastics: Theory and Applications

Suitable families of random variables having power series distributions are considered, and their asymptotic behavior in terms of large (and moderate) deviations is studied. Two examples of fractional counting processes are presented, where the normalizations of the involved power series distributions can be expressed in terms of the Prabhakar function. The first example allows to consider the counting process in [Integral Transforms Spec. Funct. 27 (2016), 783–793], the second one is inspired by a model studied in [J. Appl. Probab. 52 (2015), 18–36].

We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. This kind of processes are useful in the study of chain molecular diffusions. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subordinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in [Journal of Statistical Physics 154 (2014), 1352–1364].

The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes – the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.

We investigate large deviation properties of the maximum likelihood drift parameter estimator for Ornstein–Uhlenbeck process driven by mixed fractional Brownian motion.