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

We study the asymptotic properties of the likelihood ratio pro-cesses for a sequence of binary ¯ltered 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 ex-tend 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 super-martingales, too. Key words: asymptotic mixed normality, asymptotic normality, binary ¯l-tered experiment, contiguity, Hellinger process, likelihood ratio process, nonnegative supermartingale, weak convergence AMS-classi¯cation: 60F05, 60F17, 60G30, 60G44, 62E20