The authors study asymptotic properties of likelihood ratio processes for a sequence of binary filtered experiments by first obtaining an approximation for the log-likelihood ratio process and then applying it to obtain weak limit theorems. The limiting process is the stochastic exponential of a continuous martingale. The results obtained extend some results in Chapter 10 of [J. Jacod and A. N. Shiryaev, Limit theorems for stochastic processes, 2 nd ed. (2003; Zbl 1018.60002)].
A characterization of a certain class of exponential experiments, so-called E-experiments, is given. This allows us to give necessary and sufficient conditions for a sequence of experiments to converge to an E-experiment. The obtained results are valid for Gaussian shift experiments. Some asymptotic approximations using E-experiments are studied.