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 /\/\/\> Xnt = (Xn,ψt|ψ ∈ Ψ) where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t /\/\/\> Xn,ψt 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.
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty