### Abstract

This note extends some results of Nishiyama [Ann. Probab. 28 (2000) 685-712], A maximal inequality for stochastic integrals with respect to integer-valued random measures which may have infinitely many jumps on compact time intervals is given. By using it, a tightness criterion is obtained; if the so-called quadratic modulus is bounded in probability and if a certain entropy condition on the parameter space is satisfied, then the tightness follows. Our approach, is based on the entropy techniques developed in the modern theory of empirical processes.

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

Pages (from-to) | 1194-1200 |

Number of pages | 7 |

Journal | Annals of Probability |

Volume | 35 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2007 May |

Externally published | Yes |

### Fingerprint

### Keywords

- Entropy
- Integer-valued random
- Martingale
- Maximal inequality
- Measure
- Weak convergence

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

**On the paper "weak convergence of some classes of martingales with jumps".** / Nishiyama, Yoichi.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 35, no. 3, pp. 1194-1200. https://doi.org/10.1214/009117906000000755

}

TY - JOUR

T1 - On the paper "weak convergence of some classes of martingales with jumps"

AU - Nishiyama, Yoichi

PY - 2007/5

Y1 - 2007/5

N2 - This note extends some results of Nishiyama [Ann. Probab. 28 (2000) 685-712], A maximal inequality for stochastic integrals with respect to integer-valued random measures which may have infinitely many jumps on compact time intervals is given. By using it, a tightness criterion is obtained; if the so-called quadratic modulus is bounded in probability and if a certain entropy condition on the parameter space is satisfied, then the tightness follows. Our approach, is based on the entropy techniques developed in the modern theory of empirical processes.

AB - This note extends some results of Nishiyama [Ann. Probab. 28 (2000) 685-712], A maximal inequality for stochastic integrals with respect to integer-valued random measures which may have infinitely many jumps on compact time intervals is given. By using it, a tightness criterion is obtained; if the so-called quadratic modulus is bounded in probability and if a certain entropy condition on the parameter space is satisfied, then the tightness follows. Our approach, is based on the entropy techniques developed in the modern theory of empirical processes.

KW - Entropy

KW - Integer-valued random

KW - Martingale

KW - Maximal inequality

KW - Measure

KW - Weak convergence

UR - http://www.scopus.com/inward/record.url?scp=42649113319&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=42649113319&partnerID=8YFLogxK

U2 - 10.1214/009117906000000755

DO - 10.1214/009117906000000755

M3 - Article

AN - SCOPUS:42649113319

VL - 35

SP - 1194

EP - 1200

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 3

ER -