A Donsker-type Theorem for Log-likelihood Processes

Let $(\mathrm{\Omega },\mathcal{F},(\mathcal{F}{)}_{t\ge 0},P)$ be a complete stochastic basis, $X$ a semimartingale with predictable compensator $(B, C, \nu)$\mathrm{.}Considerafamilyofprobabilitymeasures$$\mathbf{P}$=( {P}^{n, \psi}, \psi\in \Psi, n\ge 1)$,where$\Psi$isanindexset,$ {P}^{n, \psi}\stackrel {loc} \ll{P}$,anddenotethelikelihoodratioprocessby$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 ${\mathrm{\ell }}^{\mathrm{\infty }}(\mathrm{\Psi })$ as $n\to \mathrm{\infty }$ for each fixed $t>0$.