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September 2006 A theorem on majorizing measures
Witold Bednorz
Ann. Probab. 34(5): 1771-1781 (September 2006). DOI: 10.1214/009117906000000241


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, $\mathscr{S}:=\sup_{x\in T}\int^{D(T)}_{0}\varphi^{-1}(\frac {1}{m(B(x,\varepsilon ))})\,d\varepsilon <\infty$, then $$\mathbf{E} \sup_{s,t \in T}|X(s)−X(t)|\le32\mathscr{S}$$ 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.


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Witold Bednorz. "A theorem on majorizing measures." Ann. Probab. 34 (5) 1771 - 1781, September 2006.


Published: September 2006
First available in Project Euclid: 14 November 2006

zbMATH: 1113.60040
MathSciNet: MR2271481
Digital Object Identifier: 10.1214/009117906000000241

Primary: 60G17
Secondary: 28A99

Keywords: majorizing measures , Sample boundedness

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 5 • September 2006
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