The asymptotics of multidimensional infinitely divisible distributions
The author investigates the asymptotic behavior of multidimensional infinitely divisible distributions and their densities with the support in a cone. The result of the asymptotic behavior of the distributions generalizes the results of \textit{M. S. Sgibnev} [Sib. Math. J. 31, No. 1, 115-119 (1990); translation from Sib. Mat. Zh. 31, No. 1(179), 135-140 (1990; Zbl 0703.60010)], \textit{E. Omey} [Yokohama Math. J. 33, 39-48 (1985; Zbl 0591.60012)] and \textit{S. N. Antonov} and the reviewer [Teor. Veroyatn. Primen. 27, No. 4, 625-642 (1982; Zbl 0503.60026) and ibid. 29, No. 4, 735-742 (1984; Zbl 0561.60002)]. The generalization is based on a new notion of oscillation of functions. A function \(f\) feebly oscillates at infinity in a cone \(\Gamma\) if it is defined and positive for all \(x\in\Gamma\), \(| x|\geq a>0\), and \(f(ty)/f(tx)\to 1\) as \(t\to\infty\) and \(y\to x\), \(y\in\Gamma\) for arbitrary \(x\in\Gamma\setminus\{0\}\). Quote the main result: Let \(\Gamma\subset\mathbb{R}^n\) be a cone with the vertex at zero. It is supposed that \(\Gamma\) and the adjoint cone \(\Gamma^*\) have nonempty interiors. Denote \(\Gamma(x)= \{y: y,y- x\in\Gamma\}\), \(\overline\Gamma= \mathbb{R}^n\setminus\Gamma\), \(\overline\Gamma(x)= \Gamma\setminus \Gamma(x)\), \(G= \text{interior }\Gamma\). Theorem: Let an infinitely divisible measure \(P\) on \(\Gamma\) possess the Laplace transform \[ \int_\Gamma e^{-(\lambda, x)}P(dx)= \exp\Biggl\{ \int_\Gamma (1- e^{-(\lambda, x)})Q(dx)\Biggr\},\qquad \lambda\in\Gamma^*, \] with its Lévy measure \(Q\) satisfying the conditions \(\int_{| x|\leq 1}| x| Q(dx)<\infty\), \(\int_{| x|>1}| x| Q(dx)<\infty\). If the function \(g(x)= \int_{\overline\Gamma(x)} Q(dx)\) feebly oscillates in \(G\), then \(P(\overline\Gamma(x))= (1+ o(1)) g(x)\) as \(x\to \infty\), \(x\in\Gamma_0\), where \(\Gamma_0\) is an arbitrary cone such that the closure of \(\Gamma_0\cap \{y: y\in \mathbb{R}^n,| y|= 1\}\) is in \(G\).
multidimensional infinitely divisible distributions
scientific article; zbMATH DE number 1068211
scientific article