The Annals of Probability

A theorem on majorizing measures

Witold Bednorz

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Let (T,d) be a metric space and φ:ℝ+→ℝ an increasing, convex function with φ(0)=0. We prove that if m is a probability measure m on T which is majorizing with respect to d, φ, that is, $\mathcal{S}:=\sup_{x\in T}\int^{D(T)}_{0}\varphi^{-1}(\frac {1}{m(B(x,\varepsilon ))})\,d\varepsilon \textless\infty$, then

\[\mathbf{E}\sup_{s,t\in T}|X(s)-X(t)|\leq 32\mathcal{S}\]

for each separable stochastic process X(t), tT, which satisfies $\mathbf{E}\varphi(\frac {|X(s)-X(t)|}{d(s,t)})\leq 1$ for all s, tT, st. This is a strengthening of one of the main results from Talagrand [Ann. Probab. 18 (1990) 1–49], and its proof is significantly simpler.

Article information

Ann. Probab., Volume 34, Number 5 (2006), 1771-1781.

First available in Project Euclid: 14 November 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G17: Sample path properties
Secondary: 28A99: None of the above, but in this section

Majorizing measures sample boundedness


Bednorz, Witold. A theorem on majorizing measures. Ann. Probab. 34 (2006), no. 5, 1771--1781. doi:10.1214/009117906000000241.

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