Bogachev, V. I. The asymmetry indices of stable measures. (English. Russian original) Zbl 0621.60005 Math. Notes 40, 569-575 (1986); translation from Mat. Zametki 40, No. 1, 127-138 (1986). Let \(\alpha\in (0,2]\), X be a locally convex space and \(X^*\) its dual. For every \(f\in X^*\) and every measure \(\Gamma\geq 0\) on X with bounded support let \(Q(\alpha,\Gamma,f) = tg(\pi \alpha /2)\int f| f|^{\alpha -1}d\Gamma\) for \(\alpha\neq 1\) and \(-2\pi^{-1}\int f \ln | f| d\Gamma\) for \(\alpha =1.\) The author introduces the concept of asymmetry index \(\beta(\mu)\) of a stable probability \(\mu\) on X, with exponent \(\alpha\). If \(X=R^ n\), \(\beta(\mu)\) is the infimum of all \(\beta\geq 0\) for which the characteristic function of \(\mu\) may be represented as \[ (1)\quad \phi_{\mu}(f)=\exp (if(a)-\int | f|^{\alpha}d\Gamma + i\beta Q(\alpha,\Gamma,f)), \] with an \(a\in X\) and some \(\Gamma\). For a general X, \(\beta(\mu)\) is the supremum of all \(\beta(\mu \circ T^{-1})\) with linear continuous \(T:X\to R^ n\), \(n=1,2,... \). We have \(\beta(\mu)\in [0,1].\) An open problem is: does \(\beta(\mu)\) change if we restrict to \(n=1?\) The author proves that \(\beta(\mu \circ T^{-1})\leq \beta(\mu)\) for a linear continuous T, with \(=\) if \(\ker T=0\), \(\beta(\otimes_{n\geq 1}\mu_ n) = \sup_{n}\beta(\mu_ n)\) and that \(\beta(\mu)\) is the absolute value of the ”usual coefficient” for \(X=R\) and for the distribution on the function space of a homogeneous stable process with independent increments. Every stable \(\mu\) equals \(\mu_ 1*\nu\), where \(\mu_ 1\) is the symmetrisation of \(\mu\) transported by a c.1 and \(\nu\) is stable with \(\beta(\nu)=1\) if \(\beta(\mu)>0\) and degenerated if \(\beta(\mu)=0\). c is expressed in terms of \(\beta(\mu)\) and is extremal among all for which such a representation, without supposing \(\nu\) stable, exists. If for all stable probabilities on X the representation (1) is possible with \(\beta =1\) (particularily if X is quasicomplete), then the set of all \(\beta\geq 0\) for which this is possible is \([\beta(\mu),1].\) Reviewer: I.Cuculescu Cited in 3 Documents MSC: 60B11 Probability theory on linear topological spaces 60E07 Infinitely divisible distributions; stable distributions 60E10 Characteristic functions; other transforms Keywords:stable probabilities on locally convex spaces; asymmetry index; characteristic function; stable process with independent increments PDFBibTeX XMLCite \textit{V. I. Bogachev}, Math. Notes 40, 569--575 (1986; Zbl 0621.60005); translation from Mat. Zametki 40, No. 1, 127--138 (1986) Full Text: DOI References: [1] A. Tortrat, ?Lois e(?) dans les espaces vectoriels et lois stables,? Z. Wahr. Verb. Geb.,37, No. 2, 175-182 (1976). · Zbl 0335.60013 · doi:10.1007/BF00536779 [2] N. N. Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions in Banach Spaces [in Russian], Nauka, Moscow (1985). · Zbl 0572.60003 [3] H. Shaefer, Topological Vector Spaces, Macmillan, London-New York (1966). [4] O. G. Smolyanov and S. V. Fomin, ?Measures on topological linear spaces,? Usp. Mat. Nauk,31, No. 4, 3-56 (1976). · Zbl 0364.28010 [5] V. M. Zolotarev, One-Dimensional Stable Distributions [in Russian], Nauka, Moscow (1983). · Zbl 0523.60003 [6] V. I. Averbukh, O. G. Smolyanov, and S. V. Fomin, ?Generalized functions and differential equations in linear spaces,? Tr. Mosk. Mat. Obshch.,24, 132-174 (1971). · Zbl 0234.28005 [7] R. Dudley and M. Kanter, ?Zero-one laws for stable measures,? Proc. Am. Math. Soc.,45, No. 2, 245-252 (1974). · Zbl 0297.60007 · doi:10.1090/S0002-9939-1974-0370675-9 [8] A. V. Skorokhod, Random Processes with Independent Increments [in Russian], Nauka, Moscow (1964). · Zbl 0132.12504 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.