### 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^{n}_{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.

Original language | English |
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Pages (from-to) | 459-494 |

Number of pages | 36 |

Journal | Probability Theory and Related Fields |

Volume | 108 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1997 Aug |

Externally published | Yes |

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty