NDLI: Continuity of the Multifractal Spectrum of a Random Statistically Self-Similar Measure

Content Provider	Springer Nature Link
Author	Barral, Julien
Copyright Year	2000
Abstract	Until now [see Kahane;$^{(19)}$ Holley and Waymire;$^{(16)}$ Falconer;$^{(14)}$ Olsen;$^{(29)}$ Molchan;$^{(28)}$ Arbeiter and Patzschke;$^{(1)}$ and Barral$^{(3)}$] one determines the multifractal spectrum of a statistically self-similar positive measure of the type introduced, in particular by Mandelbrot,$^{(26, 27)}$ only in the following way: let μ be such a measure, for example on the boundary of a c-ary tree equipped with the standard ultrametric distance; for α≥0, denote by E $_{α}$ the set of the points where μ possesses a local Hölder exponent equal to α, and dim E $_{α}$ the Hausdorff dimension of E $_{α}$; then, there exists a deterministic open interval I⊂ $$\mathbb{R}$$ $_{+}$ and a function f: I→ $$\mathbb{R}$$ $_{+}$ such that for all α in I, with probability one, dim E $_{α}$=f(α). This statement is not completely satisfactory. Indeed, the main result in this paper is: with probability one, for all α∈I, dim E $_{α}$=f(α). This holds also for a new type of statistically self-similar measures deduced from a result recently obtained by Liu.$^{(22)}$ We also study another problem left open in the previous works on the subject: if α=inf(I) or α=sup(I), one does not know whether E $_{α}$ is empty or not. Under suitable assumptions, we show that E $_{α}$≠ø and calculate dim E $_{α}$.
Starting Page	1027
Ending Page	1060
Page Count	34
File Format	PDF
ISSN	08949840
Journal	Journal of Theoretical Probability
Volume Number	13
Issue Number	4
e-ISSN	15729230
Language	English
Publisher	Kluwer Academic Publishers-Plenum Publishers
Publisher Date	2000-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Probability Theory and Stochastic Processes Statistics
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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