A Donsker-type Theorem for Log-likelihood Processes

Abstract

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 $Z_t^{n,\psi }=\frac{d{P}^{n,\psi }{|}_{{\mathcal{F}}_t}}{dP{|}_{{\mathcal{F}}_t}}$. 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 $n\to \infty$ for each fixed $t>0$.