  Different change-point type models encountered in statistical inference for
stochastic processes give rise to different limiting likelihood ratio
processes. In a previous paper of one of the authors it was established that
one of these likelihood ratios, which is an exponential functional of a
two-sided Poisson process driven by some parameter, can be approximated (for
sufficiently small values of the parameter) by another one, which is an
exponential functional of a two-sided Brownian motion. In this paper we
consider yet another likelihood ratio, which is the exponent of a two-sided
compound Poisson process driven by some parameter. We establish, that similarly
to the Poisson type one, the compound Poisson type likelihood ratio can be
approximated by the Brownian type one for sufficiently small values of the
parameter. We equally discuss the asymptotics for large values of the parameter
and illustrate the results by numerical simulations.
