Approximations and limit theorems for likelihood ratio processes in the binary case

We study the asymptotic properties of the likelihood ratio processes for a sequence of binary filtered experiments. First we prove approximation results for the log-likelihood ratio processes and then apply them to obtain weak limit theorems. Here the limiting process is the stochastic exponential of a continuous martingale. Our results extend the corresponding results in the well-known monograph of Jacod and Shiryaev [16, Chapter X]. It turns out that the main results are valid for nonnegative supermartingales, too.