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Isotropic sequences of random variables and stochastic rescaling. (English) Zbl 0955.60053
Theory Probab. Math. Stat. 60, 151-157 (2000) and Teor. Jmovirn. Mat. Stat. 60, 136-142 (1999).
The authors introduce the notion of $$l_{p}$$-isotropic random vectors. A random vector $$X=(X_1,\ldots,X_{n})\in R^{n}$$ is called $$l_{p}$$-isotropic if there exists a nonnegative random variable $$\Theta$$ such that the distribution of $$X$$ coincides with the distribution of $$U\cdot \Theta$$, where $$U$$ is independent of $$\Theta$$ and has a density function $$c(n,p)^{-1}(1-\sum_{i=1}^{n-1}|y_{i} |^{p})_{+}^{1/p-1}$$. A sequence of random variables $$\{ X_{i}, i=1,2,\ldots\}$$ is called $$l_{p}$$-isotropic if for every $$n\in N$$ the random vector $$X=(X_1,\ldots,X_{n})\in R^{n}$$ is $$l_{p}$$-isotropic. The authors present conditions under which a sequence of random variables is $$l_{p}$$-isotropic. For example, a sequence of random variables $$X_1,X_2,\ldots$$ is $$l_{p}$$-isotropic if there exists a probability measure $$\lambda$$ on $$[0,\infty)$$ such that for every $$k\in N$$ the density of the random vector $$X=(X_1,\ldots,X_{k})$$ has the following form $g_{k}(x_1,\ldots,x_{k})= \Biggl[{pt^{1/p}\over 2\Gamma(1/p)}\Biggr]^{k} \int_0^\infty \exp\Biggl\{ -t \sum_{i=1}^{k}|x_{i}|^{p} \Biggr\}\lambda(dt).$ The authors introduce the notion of conditionally $$l_{p}$$-isotropic sequence. They present conditions under which a random sequence is conditionally $$l_{p}$$-isotropic. They study relations between the notion of conditionally $$l_{p}$$-isotropic sequence and the notion of $$l_{p}$$-isotropic sequence.

##### MSC:
 60G60 Random fields 60G07 General theory of stochastic processes