### Abstract

Control charts based on convolutions require study of the tail behavior of the empirical distribution function of convolutions. It is well-known that this empirical distribution function at a fixed argument $x$ is asymptotically normal. The asymptotic normality is extended here to sequences $x_{n}$ tending to infinity at a suitable rate. At still larger $x_{n}$'s Poisson limiting distributions come in for the classical empirical distribution function. Surprisingly, this property does not generalize to its convolution counterpart, since for those $x_{n}$'s it is degenerate at $0$ with probability tending to 1. Exact inequalities for the tail behavior are presented as well.

Original language | Undefined |
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Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 25 |

Publication status | Published - 2004 |

### Publication series

Name | Memorandum Faculty of Applied Mathematics |
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Publisher | Department of Applied Mathematics, University of Twente |

No. | 1740 |

ISSN (Print) | 0169-2690 |

### Keywords

- MSC-62E20
- EWI-3560
- IR-65924
- MSC-62P30
- METIS-219690
- MSC-62G30

## Cite this

Albers, WW., & Kallenberg, W. C. M. (2004).

*Tail behavior of the empirical distribution function of convolutions*. (Memorandum Faculty of Applied Mathematics; No. 1740). Enschede: University of Twente, Department of Applied Mathematics.