### Abstract

This paper deals with weak convergence of stochastic integrals with respect to multivariate point processes. The results are given in terms of an entropy condition for partitioning of the index set of the integrands, which is a sort of L2-bracketing. We also consider lα-valued martingale difference arrays, and present natural generalizations of Jain-Marcus's and Ossian-der's central limit theorems. As an application, the asymptotic behavior of log-likelihood ratio random fields in general statistical experiments with abstract parameters is derived.

Original language | English |
---|---|

Pages (from-to) | 685-712 |

Number of pages | 28 |

Journal | Annals of Probability |

Volume | 28 |

Issue number | 2 |

Publication status | Published - 2000 Apr |

Externally published | Yes |

### Fingerprint

### Keywords

- Central limit theorem
- Likelihood
- Markov chain
- Martingale
- Point process
- Weak convergence

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Annals of Probability*,

*28*(2), 685-712.

**Weak convergence of some classes of martingales with jumps.** / Nishiyama, Yoichi.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 28, no. 2, pp. 685-712.

}

TY - JOUR

T1 - Weak convergence of some classes of martingales with jumps

AU - Nishiyama, Yoichi

PY - 2000/4

Y1 - 2000/4

N2 - This paper deals with weak convergence of stochastic integrals with respect to multivariate point processes. The results are given in terms of an entropy condition for partitioning of the index set of the integrands, which is a sort of L2-bracketing. We also consider lα-valued martingale difference arrays, and present natural generalizations of Jain-Marcus's and Ossian-der's central limit theorems. As an application, the asymptotic behavior of log-likelihood ratio random fields in general statistical experiments with abstract parameters is derived.

AB - This paper deals with weak convergence of stochastic integrals with respect to multivariate point processes. The results are given in terms of an entropy condition for partitioning of the index set of the integrands, which is a sort of L2-bracketing. We also consider lα-valued martingale difference arrays, and present natural generalizations of Jain-Marcus's and Ossian-der's central limit theorems. As an application, the asymptotic behavior of log-likelihood ratio random fields in general statistical experiments with abstract parameters is derived.

KW - Central limit theorem

KW - Likelihood

KW - Markov chain

KW - Martingale

KW - Point process

KW - Weak convergence

UR - http://www.scopus.com/inward/record.url?scp=0034345594&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034345594&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034345594

VL - 28

SP - 685

EP - 712

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 2

ER -