On small deviations of Gaussian processes using majorizing measures

Michel J. G. Weber¹

¹ Université Louis-Pasteur et C.N.R.S. 7 rue René Descartes 67084 Strasbourg Cedex, France

Colloquium Mathematicum, Tome 129 (2012) no. 1, pp. 41-59.

Résumé

We give two examples of periodic Gaussian processes, having entropy numbers of exactly the same order but radically different small deviations. Our construction is based on Knopp's classical result yielding existence of continuous nowhere differentiable functions, and more precisely on Loud's functions. We also obtain a general lower bound for small deviations using the majorizing measure method. We show by examples that our bound is sharp. We also apply it to Gaussian independent sequences and to a generic class of ultrametric Gaussian processes.

Détail

DOI : 10.4064/cm129-1-3

Keywords: examples periodic gaussian processes having entropy numbers exactly order radically different small deviations construction based knopps classical result yielding existence continuous nowhere differentiable functions precisely louds functions obtain general lower bound small deviations using majorizing measure method examples bound sharp apply gaussian independent sequences generic class ultrametric gaussian processes