Majorizing measures on metric spaces

The majorizing measure approach to sample boundedness

“…Note that in the case of Gaussian processes the above property was proved in [1], Theorem 4.2, yet also mentioned in [7] and known to Talagrand [15]. There are many cases (see [3,4,5]) where one can prove the lower bound on the supremum of stochastic processes in the form sup µ M(µ, µ). The benefit of the approach is that the lower bound has to be found for a given measure µ on T , which better fits the chaining argument.…”

“…In this section we collect all the upper bounds required in this paper. The basic theory was given in [16] and then slightly developed in [2] and [3]. First note that our measure approach works in much generalized setting.…”

On the convergence of orthogonal series

“…Then appeared the characterization of sample boundedness for Gaussian processes [9] and many other canonical processes [11,5]. Also, the author could generalize the result for the ultrametric spaces to a setting [2] which in the special suborthogonal case gives: Theorem 5. Whenever each process X(t), t ∈ T that satisfies (2) is sample bounded then there exists a majorizing measure on T .…”

Integrability and concentration of the truncated variation for the sample paths of fractional Brownian motions, diffusions and Lévy processes

“…The chaining approach was first used to study problems of sample boundedness of processes on the general index space Fernique [4,5]. The method was developed to give the full description of classes of processes that are sample bounded, under certain integrability condition Bednorz [1,2], Bednorz [3], Ledoux and Talagrand [8], Talagrand [17], and the small ball probability Li and Shao [9]. For a comprehensive study where many analytical examples are given, see Talagrand [18].…”