| Abstract: |
| Let $(\Omega, \mathcal{F}, (\mathcal{F})_{t\ge 0}, P)$ be a complete stochastic basis, $X$ a semimartinagle with predictable compensator $(B, C, \nu)$. Consider a family of probability measures $\mathbf{P}=( {P}^{n, \psi}, \psi\in \Psi)$, $n\ge 1$, where $ {P}^{n, \psi}\stackrel {loc} \ll{P}$, and denote the likelihood ratio process by $Z_t^{n, \psi} =\frac{d\, P^{n, \psi}|_{\mathcal{F}_t}}{d\, P|_{\mathcal{F}_t}}$. We are mainly interested in the logarithmic ratio process $\log Z_t^{n, \psi}$. Under some regularity conditions in terms of logarithm entropy and Hellinger processes, we prove that $\log Z_t^{n}$ converges weakly to a Gaussian process in $\ell^\infty(\Psi)$ as $n\rightarrow\infty$. At last, an application of our main result is given. |
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