### 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 |
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Pages (from-to) | 1194-1200 |

Number of pages | 7 |

Journal | Annals of Probability |

Volume | 35 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2007 May 1 |

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### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty