Large deviations for sums of random vectors attracted to operator semi-stable laws
type: Item
instance of: scholarly article
title: Large deviations for sums of random vectors attracted to operator semi-stable laws
zbMATH Open document ID: 1368.60029
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Journal of Theoretical Probability
publication date: 2017-04-12T00:00:00Z
review text: Let \(X, X_1, X_2, \dots\) be i.i.d. \(\mathbb{R}^d\)-valued random vectors, \(d \geq 1,\) \(S_n = X_1 + \cdots + X_n,\) and \(Y\) be a random vector without normal component whose distribution is full in the sense that it cannot be supported on any \((d-1)\)-dimensional hyperplane. \(X\) is said to belong to the strict generalized domain of operator semi-stable attraction, denoted by \(X \in \text{GDOSA}(Y,c),\) if there exist an increasing sequence \(\{k(n)\}\) of natural numbers tending to \(\infty,\) with \(k(n+1)/k(n)\) tending to a constant \(c > 1,\) and a regularly varying sequence \(\{B_n\}\) of linear operators with \(B_{\lambda n}B_n^{-1}\) tending to \(\lambda^{-E} = \exp(-E \log \lambda) = \sum_{j=0}^{\infty} \frac{(-E \log \lambda)^j}{j!}\) for all \(\lambda > 0,\) where \(E\) is an invertible \(d \times d\) matrix, such that \(B_{k(n)}S_{k(n)}\) converges in distribution to \(Y\). After proving some preliminary results, the first main result shows that if \(X \in \text{GDOSA}(Y,c),\) then for all unit vectors \(\theta \in \mathbb{R}^d,\) \(P(|\langle S_n, \theta \rangle| > x ) \sim n P( |\langle X, \theta \rangle| > x )\) holds uniformly for \(x \geq \nu_n,\) where \(\{\nu_n\}\) is a monotone sequence of positive numbers satisfying some conditions and \(\langle\;,\;\rangle\) denotes the usual Euclidean inner-product in \(\mathbb{R}^d.\) The second main result shows that if \(X \in \text{GDOSA}(Y,c)\) and \(\{N(t), t \geq 0\}\) is a nonnegative and integer-valued sequence independent of \(\{X_n, n \geq 1\},\) with \(\lambda(t) = E(N(t))\) tending to \(\infty\) and \(E(N(t) I(N(t) > (1 + \varepsilon) \lambda(t))) = o(\lambda(t))\) for \(\varepsilon > 0,\) then \(P(|\langle S_{N(t)}, \theta \rangle| > x ) \sim \lambda(t) \sim P( |\langle X, \theta \rangle| > x )\) holds uniformly for \(x \geq \nu(t),\) where \(\{\nu(t)\}\) is a nondecreasing and nonnegative integer-valued process satisfying some conditions. The proofs make use of the theory of multivaria…
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60K40
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 6705453
zbMATH Keywords: large deviations
zbMATH Keywords: random vectors
zbMATH Keywords: operator semi-stable law
zbMATH Keywords: domain of attraction
zbMATH Keywords: multivariate regular variation
zbMATH Keywords: heavy tails
reviewed by: Sreenivasan Ravi
MaRDI profile type: Publication
full work available at URL: s10959-015-0645-5
OpenAlex ID: W1917020743
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DOI: 10.1007/S10959-015-0645-5
description: scientific article; zbMATH DE number 6705453