Large deviation results on some estimators for stationary Gaussian processes

Abstract

In this paper, we present large deviation results for estimators of some unknown parameters concerning stationary Gaussian processes. We deal with both maximum likelihood estimators and posterior distributions; moreover, we illustrate the differences between short-range- and long-range-dependent processes. As a typical feature the rate functions for maximum likelihood estimators and posterior distributions are given in terms of the same relative entropy and the roles of the two probability measures in the relative entropy are exchanged. We define a sort of relative entropy with respect to the sampling process which in the i.i.d. case corresponds to the relative entropy with respect to the common law of each single sample. In view of potential applications in risk theory we prove large deviation results for estimators of the logarithmic asymptotic decay rate of the tail of the supremum of a random walk with stationary Gaussian increments. Finally, we present results for compound renewal processes with stationary Gaussian distributed rewards, independent of i.i.d. Weibull distributed renewal times.

Keywords:

• large deviations
• stationary Gaussian process
• spectral density
• short-range and long-range dependence
• level crossing probability
• compound renewal process

2000 Mathematical Subject Classification :

• 60F10
• 60G10
• 60G15
• 62F10
• 62F12
• 62F15

Acknowledgements

We thank Domenico Marinucci for several useful discussions and comments, and Ken Duffy for the illustration of the results in the literature on semi-exponential distributions. We also thank two anonymous referees for their useful comments. This work has been partially supported by Murst Project Metodi Stocastici in Finanza Matematica.