Large deviations for sums of random vectors attracted to operator semi-stable laws
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 multivariate regular variation. The results are applied to random variables with regularly varying tails and to weakly negatively associated random variables.
scientific article; zbMATH DE number 6705453