Some central limit theorems for ℓ^∞-valued semimartingales and their applications

This paper is devoted to the generalization of central limit theorems for empirical processes to several types of ℓ^∞(Ψ)-valued continuous-time stochastic processes t /\/\/\> Xⁿ_t = (X^n,ψ_t|ψ ∈ Ψ) where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t /\/\/\> X^n,ψ_t is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived.