Stationarity, isotropy and sphericity in ℓp

  • Simeon M. Berman


Let Xn,−∞<n<∞ be a stationary sequence of random variables. For each n>0, and fixed p>0 put Rn=(¦X1¦p+...+¦Xn¦p)1/p and assume P(Rn=0)=0. If for each n, the random point (X1/Rn,..., Xn/Rn) is uniformly distributed on the unit n-dimensional ℝp-sphere, then the random variables X1,..., Xd have a joint density of the form
$$\mathop \smallint \limits_0^\infty y^{{d \mathord{\left/ {\vphantom {d p}} \right. \kern-\nulldelimiterspace} p}} (2\Gamma ({1 \mathord{\left/ {\vphantom {1 {p + 1)p}}} \right. \kern-\nulldelimiterspace} {p + 1)p}}^{1/p} )^{ - d} \exp \left( { - \frac{y}{p}\sum\limits_{i = 1}^d {|x_i |p} } \right)dG(y),$$
for some distribution G on (0, ∞), for every d>0. This property is called sphericity.


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  1. 1.
    Berman, S.M.: Second order random fields over p with homogeneous and isotropic increments. Z. Wahrscheinlichkeitstheorie verw. Geb. 12, 107–126 (1969)Google Scholar
  2. 2.
    Letac, G.: Isotropy and sphericity, some characterizations of the normal distribution. Submitted to Ann. Statist.Google Scholar
  3. 3.
    Letac, G., Milhaud, X.: Une suite stationnaire et isotrope est spherique. Z. Wahrscheinlichkeitstheorie verw. Geb. 49, 33–36 (1979)Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Simeon M. Berman
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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