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Different change-point type models encountered in statistical inference forstochastic processes give rise to different limiting likelihood ratioprocesses. In this paper we consider two such likelihood ratios. The first oneis an exponential functional of a two-sided Poisson process driven by someparameter, while the second one is an exponential functional of a two-sidedBrownian motion. We establish that for sufficiently small values of theparameter, the Poisson type likelihood ratio can be approximated by theBrownian type one. As a consequence, several statistically interestingquantities (such as limiting variances of different estimators) related to thefirst likelihood ratio can also be approximated by those related to the secondone. Finally, we discuss the asymptotics of the large values of the parameterand illustrate the results by numerical simulations.

On Limiting Likelihood Ratio Processes of some Change-Point Type Statistical Models