### 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 |
---|---|

Pages (from-to) | 459-494 |

Number of pages | 36 |

Journal | Probability Theory and Related Fields |

Volume | 108 |

Issue number | 4 |

Publication status | Published - 1997 Aug |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Statistics and Probability

### Cite this

**Some central limit theorems for ℓ ^{∞}-valued semimartingales and their applications.** / Nishiyama, Yoichi.

Research output: Contribution to journal › Article

^{∞}-valued semimartingales and their applications',

*Probability Theory and Related Fields*, vol. 108, no. 4, pp. 459-494.

}

TY - JOUR

T1 - Some central limit theorems for ℓ∞-valued semimartingales and their applications

AU - Nishiyama, Yoichi

PY - 1997/8

Y1 - 1997/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0002154328&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002154328&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0002154328

VL - 108

SP - 459

EP - 494

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 4

ER -