A remark on normalizations in a local large deviations principle for inhomogeneous birth – and – death process

This work is a continuation of [13]. We consider a continuous-time birth – and – death process in which the transition rates are regularly varying function of the process position. 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.