A Donsker-Type Theorem for Log-Likelihood Processes

Zhonggen Su () and Hanchao Wang ()
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Zhonggen Su: Zhejiang University
Hanchao Wang: Shandong University

Journal of Theoretical Probability, 2020, vol. 33, issue 3, 1401-1425

Abstract: Abstract Let $$(\Omega , \mathcal {F}, (\mathcal {F})_{t\ge 0}, P)$$ ( Ω , F , ( F ) t ≥ 0 , P ) be a complete stochastic basis, and X be a semimartingale with predictable compensator $$(B, C, \nu )$$ ( B , C , ν ) . Consider a family of probability measures $$\mathbf {P}=( {P}^{n, \psi }, \psi \in \Psi , n\ge 1)$$ P = ( P n , ψ , ψ ∈ Ψ , n ≥ 1 ) , where $$\Psi $$ Ψ is an index set, $$ {P}^{n, \psi }{\mathop {\ll }\limits ^\mathrm{loc}}{P}$$ P n , ψ ≪ loc P , and denote the likelihood ratio process by $$Z_t^{n, \psi } =\frac{\mathrm{d}P^{n, \psi }|_{\mathcal {F}_t}}{\mathrm{d} P|_{\mathcal {F}_t}}$$ Z t n , ψ = d P n , ψ | F t d P | F t . Under some regularity conditions in terms of logarithm entropy and Hellinger processes, we prove that $$\log Z_t^{n}$$ log Z t n converges weakly to a Gaussian process in $$\ell ^\infty (\Psi )$$ ℓ ∞ ( Ψ ) as $$n\rightarrow \infty $$ n → ∞ for each fixed $$t>0$$ t > 0 .

Keywords: Hellinger process of order zero; Log-likelihood process; Semimartinagle; Weak convergence; 60F05; 60F17 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10959-019-00926-9

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