Small values and functional laws of the iterated logarithm for operator fractional Brownian motion

Abstract

The multivariate Gaussian random fields with matrix-based scaling laws are widely used for inference in statistics and many applied areas. In such contexts, interests are often Hölder regularities of spatial surfaces in any given direction. This article analyzes the almost sure sample function behavior for operator fractional Brownian motion, including multivariate fractional Brownian motion. We obtain the estimations of small ball probability and the strongly locally nondeterministic for operator fractional Brownian motion in any given direction. By applying these estimates, we obtain Chung type laws of the iterated logarithm for operator fractional Brownian motion. Our results show that the precise Hölder regularities of these spatial surfaces are completely determined by the real parts of the eigenvalues of self-similarity exponent and the covariance matrix at time point 1.

Keywords

• operator self-similarity
• operator fractional brownian motion
• small ball probability
• chung type laws of the iterated logarithm
• strong local nondeterminism
• 60f15
• 60g17
• 60g22
• 60g60