Let \(X\) be a symmetric random variable and \(X_1\), \(X_2\) be independent copies of \(X\). \(X\) is called \(p\)-stable if for every real numbers \(a\), \(b\) the variable \(a X_1 + b X_2\) has the same distribution as \((| a| ^p + | b| ^p)^{1/p} X\). It is well known that \(p\)-stable variables exist for \(0<p\leq 2\) (\(p=2\) corresponds to the case of Gaussian random variables) and do not exist for \(p>2\), and that the characteristic function of a such variable is \(\exp(-c | t| ^p)\) for some positive constant \(c\).
The author introduces \(p\)-pseudostable random variables for \(p>2\). Namely, given \(p>2\), we say that \(X\) is \(p\)-pseudostable if for every real numbers \(a\), \(b\) there exists a number \(v=v(a, b)\) such that the variable \(a X_1 + b X_2\) has the same distribution as \((| a| ^p + | b| ^p)^{1/p} X + v G\), where \(G\) is standard Gaussian random variable. First the author proves that \(p\)-pseudostable random variables exist if and only if \(4k < p < 4k +2\) for some integer \(k\geq 1\); shows that the characteristic function of a such variable is \(\exp(-c | t| ^p -\alpha t^2)\) for some positive constants \(c\) and \(\alpha\); and investigates some other properties of such variables. Then he provides applications to the embedding of so-called Rosenthal type spaces, i.e. spaces of square summable sequences equipped with the norm \(\| a\| = \| a\| _p + s \| a\| _2\) for some \(s>0\), into \(L_q\) for \(1\leq q \leq p\). Finally, he applies \(p\)-pseudostable random variables to compare volumes of central \((n-1)\)-dimensional sections \(H_u\) and \(H_v\) of the unit ball \(B_p\) of \(\ell _p^n\) orthogonal to vectors \(u=(1, 1, 1, \dots, 1)\) and \(v=(1, 1, 0, \dots, 0)\), respectively. More precisely, it is shown that the limit of the ratio of volumes of \(H_u\) and \(H_v\) as \(n\) grows to infinity is smaller than \(1\) for large \(p\) and greater than \(1\) for small \(p\). Recall that it is still unknown how the largest central section of \(B_p\) looks like. In particular, is it true that \(H_v\) has the largest volume for large \(p\) [as it is for \(p=\infty\), see K. Ball, Proc. Am. Math. Soc. 97, 465–473 (1986; Zbl 0601.52005)]?
For the entire collection see [Zbl 1009.00019].