Positive definite functions and multidimensional versions of random variables
| dc.creator | Koldobsky, Alexander | |
| dc.date | 2009-03-08 | |
| dc.date.accessioned | 2026-07-07T12:50:22Z | |
| dc.date.available | 2026-07-07T12:50:22Z | |
| dc.description | We say that a random vector $X=(X_1,...,X_n)$ in $R^n$ is an $n$-dimensional version of a random variable $Y$ if for any $a\in R^n$ the random variables $\sum a_iX_i$ and $γ(a) Y$ are identically distributed, where $γ:R^n\to [0,\infty)$ is called the standard of $X.$ An old problem is to characterize those functions $γ$ that can appear as the standard of an $n$-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in $L_0.$ This result is almost optimal, as the norm of any finite dimensional subspace of $L_p$ with $p\in (0,2]$ is the standard of an $n$-dimensional version ($p$-stable random vector) by the classical result of P.Lèvy. An equivalent formulation is that if a function of the form $f(\|\cdot\|_K)$ is positive definite on $R^n,$ where $K$ is an origin symmetric star body in $R^n$ and $f:R\to R$ is an even continuous function, then either the space $(R^n,\|\cdot\|_K)$ embeds in $L_0$ or $f$ is a constant function. Combined with known facts about embedding in $L_0,$ this result leads to several generalizations of the solution of Schoenberg's problem on positive definite functions. | |
| dc.identifier | https://arxiv.org/abs/0903.1433 | |
| dc.identifier | http://arxiv.org/abs/0903.1433 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/222666 | |
| dc.subject | Probability | |
| dc.subject | Functional Analysis | |
| dc.subject | 60E05 | |
| dc.title | Positive definite functions and multidimensional versions of random variables | |
| dc.type | text |