AbstractLet $$(\Omega , \mathcal {F}, (\mathcal {F})_{t\ge 0}, P)$$(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=(Pn,ψ,ψ∈Ψ,n≥1), where $$\Psi $$Ψ is an index set, $$ {P}^{n, \psi }{\mathop {\ll }\limits ^\mathrm{loc}}{P}$$Pn,ψ≪locP, and denote the likelihood ratio process by $$Z_t^{n, \psi } =\frac{\mathrm{d}P^{n, \psi }|_{\mathcal {F}_t}}{\mathrm{d} P|_{\mathcal {F}_t}}$$Ztn,ψ=dPn,ψ|FtdP|Ft. Under some regularity conditions in terms of logarithm entropy and Hellinger processes, we prove that $$\log Z_t^{n}$$logZtn converges weakly to a Gaussian process in $$\ell ^\infty (\Psi )$$ℓ∞(Ψ) as $$n\rightarrow \infty $$n→∞ for each fixed $$t>0$$t>0.

中文翻译：

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对数似然过程的 Donsker 型定理

摘要设 $$(\Omega , \mathcal {F}, (\mathcal {F})_{t\ge 0}, P)$$(Ω,F,(F)t≥0,P) 是一个完全随机变量基，X 是一个半鞅，具有可预测的补偿器 $$(B, C, \nu )$$(B,C,ν)。考虑一系列概率测度 $$\mathbf {P}=( {P}^{n, \psi }, \psi \in \Psi , n\ge 1)$$P=(Pn,ψ,ψ∈Ψ ,n≥1), 其中 $$\Psi $$Ψ 是一个索引集, $$ {P}^{n, \psi }{\mathop {\ll }\limits ^\mathrm{loc}}{P} $$Pn,ψ≪locP，用$$Z_t^{n, \psi } =\frac{\mathrm{d}P^{n, \psi }|_{\mathcal {F}表示似然比过程_t}}{\mathrm{d} P|_{\mathcal {F}_t}}$$Ztn,ψ=dPn,ψ|FtdP|Ft。在对数熵和 Hellinger 过程的一些正则条件下，我们证明 $$\log Z_t^{n}$$logZtn 弱收敛到 $$\ell ^\infty (\Psi )$$ℓ∞ 中的高斯过程(Ψ) 为 $$n\rightarrow \infty $$n→∞ 对于每个固定的 $$t>0$$t>0。