Statistica Sinica 24 (2014), 957-969

EXACT MODERATE AND LARGE DEVIATIONS

FOR LINEAR PROCESSES

Magda Peligrad^{1}, Hailin Sang^{2}, Yunda Zhong^{3} and Wei Biao Wu^{3}

and ^{3}University of Chicago

Abstract: Large and moderate deviation probabilities play an important role in many applied areas, such as insurance and risk analysis. This paper studies the exact moderate, and large deviation asymptotics in non-logarithmic form for linear processes with independent innovations. The linear processes we analyze are general and they include the long memory case. We give an asymptotic representation for the probability of the tail of the normalized sums and specify the zones in which it can be approximated either by a standard normal distribution or by the marginal distribution of the innovation process. The results are then applied to regression estimates, moving averages, fractionally integrated processes, linear processes with regularly varying exponents, and functions of linear processes. We also consider the computation of value at risk and expected shortfall, fundamental quantities in risk theory and finance.

Key words and phrases: Large deviation, linear process, long memory, moderate deviation, non-logarithmic asymptotics, zone of normal convergence.