Stationarity, isotropy and sphericity in ℓp

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Summary

Let X n,−∞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 X 1,..., X d 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.