## Abstract

Let Z = {hellip;, - 1, 0, 1, ...}, ξ, ξ_{n}, n ε{lunate} Z a doubly infinite sequence of i.i.d. random variables in a separable Banach space B, and a_{n}, n ε{lunate} Z, a doubly infinite sequence of real numbers with 0 ≠ ∑_{n ε{lunate} z}|a_{n}| < ∞. Set X_{n} = ∑_{iε{lunate}z}a_{i}ξ_{i + n}, n ≥ 1. In this article, we prove that (X_{1} + X_{2} + ... + X_{n}) n, n ≥ 1 satisfies the upper bound of the large deviation principle if and only if E exp q_{k}(ξ) < ∞, for some compact subset K of B, where q_{k}(·) is the Minkowski functional of the set K. Interestingly enough, however, the lower bound holds without any conditions at all! We will also present an asymptotic property of the corresponding rate function.

Original language | English (US) |
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Pages (from-to) | 309-320 |

Number of pages | 12 |

Journal | Stochastic Processes and their Applications |

Volume | 59 |

Issue number | 2 |

DOIs | |

State | Published - 1995 |

### Bibliographical note

Funding Information:Jiang and Wang gratefully acknowledges the partial support of National Natural Science Foundation of China for their research work in this paper. Rao's work was supported in part by the US Army Research Office under Grant # DAAH04-93-G-0030 and NSF-EPSCoR grant. Jiang is currently with the Department of Statistics at North Dakota State University. The authors are extremely grateful to the referee and the associate editor in charge of the paper for their perceptive comments which led to a substantial improvement in the delineation of the results of the paper. Special thanks go to the associate editor for his meticulous reading of the paper.

## Keywords

- Large deviations
- Moving average processes
- Rate functions
- Truncation