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August, 1975 Stable Densities Under Change of Scale and Total Variation Inequalities
Marek Kanter
Ann. Probab. 3(4): 697-707 (August, 1975). DOI: 10.1214/aop/1176996309


In this paper it is shown that if $q$ is the density of a symmetric stable density, then for $c \in (0, 1) \cup (1, \infty)$, the graph of $q(x)$ intersects the graph of $cq(cx)$ at only two points. The argument proceeds by introducing a new characterization of unimodality for densities and involves a representation for symmetric stable random variables that is also useful for simulating such random variables. Finally our results are applied to prove some inequalities concerning the total variation norm of the difference of two symmetric stable densities.


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Marek Kanter. "Stable Densities Under Change of Scale and Total Variation Inequalities." Ann. Probab. 3 (4) 697 - 707, August, 1975.


Published: August, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0323.60013
MathSciNet: MR436265
Digital Object Identifier: 10.1214/aop/1176996309

Primary: 60E05
Secondary: 60D05

Keywords: monotone likelihood ratio , stable density , total variation distance , totally positive kernel , Unimodal density

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 4 • August, 1975
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