A Donsker-Type Theorem for Log-Likelihood Processes

  • Zhonggen Su
  • Hanchao WangEmail author


Let \((\Omega , \mathcal {F}, (\mathcal {F})_{t\ge 0}, P)\) be a complete stochastic basis, and X be 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, \( {P}^{n, \psi }{\mathop {\ll }\limits ^\mathrm{loc}}{P}\), and denote the likelihood ratio process by \(Z_t^{n, \psi } =\frac{\mathrm{d}P^{n, \psi }|_{\mathcal {F}_t}}{\mathrm{d} P|_{\mathcal {F}_t}}\). 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 \(t>0\).


Hellinger process of order zero Log-likelihood process Semimartinagle Weak convergence 

Mathematics Subject Classification (2010)

60F05 60F17 



The authors would like to thank the anonymous referees and the Associate Editor for careful reading and constructive comments. This work was supported by the National Natural Science Foundation of China (Nos. 11371317, 11701331, 11731012, 11871425) , Fundamental Research Funds for Central Universities, Shandong Provincial Natural Science Foundation (No. ZR2017QA007) and Young Scholars Program of Shandong University.


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Authors and Affiliations

  1. 1.School of Mathematical ScienceZhejiang UniversityHangzhouChina
  2. 2.Zhongtai Securities Institute for Financial StudiesShandong UniversityJinanChina

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