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2006/2007 Information Parameters and Large Deviation Spectrum of Discontinuous Measures
Julien Barral, Stéphane Seuret
Real Anal. Exchange 32(2): 429-454 (2006/2007).


Let $\nu$ be a finite Borel measure on $[0,1]^d$. Consider the $L^q$-spectrum of $\nu$: $\tau_\nu(q)=\liminf_{n\to\infty}-n^{-1} \log_b\sum_{Q\in\mathcal{G}_n}\nu (Q)^q$ ($q\ge 0$), where $\mathcal{G}_n$ is the set of $b$-adic cubes of generation $n$ ($b$ integer $\ge 2$). Let $q_\tau=\inf\{q\: \tau_\nu(q)=0\}$ and $H_\tau=\tau_\nu'(q_\tau^-)$. When $\nu$ is a mono-dimensional continuous measure of information dimension $D$, $(q_\tau,H_\tau)=(1,D)$. When $\nu$ is purely discontinuous, its information dimension is $D=0$, but the pair $(q_\tau,H_\tau)$ may be non-trivial and contains relevant information on the distribution of $\nu$. Intrinsic characterizations of $(q_\tau,H_\tau)$ are found, as well as sharp estimates for the large deviation spectrum of $\nu$ on $[0,H_\tau]$. We exhibit the differences between the cases $q_\tau=1$ and $q_\tau\in (0,1)$. We conclude that the large deviation spectrum's properties observed for specific classes of measures are true in general.


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Julien Barral. Stéphane Seuret. "Information Parameters and Large Deviation Spectrum of Discontinuous Measures." Real Anal. Exchange 32 (2) 429 - 454, 2006/2007.


Published: 2006/2007
First available in Project Euclid: 3 January 2008

zbMATH: 1129.60028
MathSciNet: MR2369854

Primary: 28A80 , 60F10

Keywords: Fractal , large deviations

Rights: Copyright © 2006 Michigan State University Press

Vol.32 • No. 2 • 2006/2007
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