A theorem on majorizing measures

Abstract

Let $(T,d)$ be a metric space and $\varphi :{\mathbb{R}}_{+}\to \mathbb{R}$ an increasing, convex function with $\varphi (0)=0$. We prove that if $m$ is a probability measure $m$ on $T$ which is majorizing with respect to $d,\varphi$, that is, $\mathcal{S}:={\sup }_{x\in T}{\int }_0^{D(T)}{\varphi }^{-1}(\frac{1}{m(B(x,\epsilon ))})d\epsilon <\infty$, then \[\mathbf{E}\sup_{s,t\in T}|X(s)-X(t)|\leq 32\mathcal{S}\] for each separable stochastic process $X(t)$, $t\in T$, which satisfies $\mathbf{E}\varphi (\frac{|X(s)-X(t)|}{d(s,t)})\le 1$ for all $s,t\in T$, $s\ne t$. This is a strengthening of one of the main results from Talagrand [Ann. Probab. 18 (1990) 1--49], and its proof is significantly simpler.

Published at http://dx.doi.org/10.1214/009117906000000241 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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