# A Donsker-type Theorem for Log-likelihood Processes

## TL;DR

This paper establishes a Donsker-type weak convergence result for the log-likelihood processes of a family of probability measures associated with a semimartingale, showing convergence to a Gaussian process under certain regularity conditions.

## Contribution

It introduces a novel weak convergence theorem for log-likelihood processes in a semimartingale setting, extending classical results to more general stochastic processes.

## Key findings

- Log-likelihood processes converge weakly to a Gaussian process
- Convergence holds under conditions involving entropy and Hellinger processes
- Results apply to a broad class of semimartingales

## 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 $Z_t^{n, \psi} =\frac{dP^{n, \psi}|_{\mathcal{F}_t}}{d P|_{\mathcal{F}_t}}$. 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 $t>0$.