A Donsker-type Theorem for Log-likelihood Processes

Let $(\Omega, \mathcal{F}, (\mathcal{F})_{t\ge 0}, P)$ be a complete stochastic basis, $X$ a semimartingale with predictable compensator $(B, C, \nu)$. Consider a family of probability measures $\mathbf{P}=( {P}^{n, \psi}, \psi\in \Psi, n\ge 1)$, where $\Psi$ is an index set, $ {P}^{n, \psi}\stackrel {loc} \ll{P}$, and denote the likelihood ratio process by $Z_t^{n, \psi} =\frac{dP^{n, \psi}|_{\mathcal{F}_t}}{d P|_{\mathcal{F}_t}}$. 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 $n\rightarrow\infty$ for each fixed $t>0$.