A theorem on majorizing measures

Details A theorem on majorizing measures A theorem on majorizing measures

2005-10-18

arxiv.org/abs/math/0510373v2

Annals of Probability 2006, Vol. 34, No. 5, 1771-1781

Mathematics

Probability

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

Scientific paper

10.1214/009117906000000241

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

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