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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 Galton–Watson process is the simplest example of a branching process. The relationship between the offspring distribution, and, when the extinction occurs almost surely, the distribution of the total progeny is well known. In this paper, we illustrate the relationship between these two distributions when we consider the large deviation rate function (provided by Cramér’s theorem) for empirical means of i.i.d. random variables. We also consider the case with a random initial population. In the final part, we present large deviation results for sequences of estimators of the offspring mean based on i.i.d. replications of total progeny.

The Galton–Watson process is the simplest example of a branching process. The relationship between the offspring distribution, and, when the extinction occurs almost surely, the distribution of the total progeny is well known. In this paper, we illustrate the relationship between these two distributions when we consider the large deviation rate function (provided by Cramér’s theorem) for empirical means of i.i.d. random variables. We also consider the case with a random initial population. In the final part, we present large deviation results for sequences of estimators of the offspring mean based on i.i.d. replications of total progeny.