×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite