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

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

Original languageEnglish
Pages (from-to)459-494
Number of pages36
JournalProbability Theory and Related Fields
Volume108
Issue number4
Publication statusPublished - 1997 Aug
Externally publishedYes

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ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Statistics and Probability

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