Stationarity, isotropy and sphericity in ℓp

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Let X n,−∞<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 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.

This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Science Foundation, Grant MCS 79-02020