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

Abstract

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⇝X t n =(X t n ,ψ|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X t n ,ψ 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.