Large deviations for moving average processes

Tiefeng Jiang, M. Bhaskara Rao, Xiangchen Wang

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6 Scopus citations


Let Z = {hellip;, - 1, 0, 1, ...}, ξ, ξn, n ε{lunate} Z a doubly infinite sequence of i.i.d. random variables in a separable Banach space B, and an, n ε{lunate} Z, a doubly infinite sequence of real numbers with 0 ≠ ∑n ε{lunate} z|an| < ∞. Set Xn = ∑iε{lunate}zaiξi + n, n ≥ 1. In this article, we prove that (X1 + X2 + ... + Xn) n, n ≥ 1 satisfies the upper bound of the large deviation principle if and only if E exp qk(ξ) < ∞, for some compact subset K of B, where qk(·) 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 languageEnglish (US)
Pages (from-to)309-320
Number of pages12
JournalStochastic Processes and their Applications
Issue number2
StatePublished - 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.


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


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