Kleffe, J.; Thrum, R. Inequalities for moments of quadratic forms with applications to a.s. convergence. (English) Zbl 0545.60027 Math. Operationsforsch. Stat., Ser. Stat. 14, 211-216 (1983). Let \(\epsilon\) be a random vector with \(E\epsilon =0\) and \(E\epsilon \epsilon^ T=\Sigma\). Denote by \({\mathfrak A}=\{A\}\) the class of all symmetric matrices such that \(\epsilon^ TA\epsilon\) is well defined, \(v(\epsilon,A)=var(\epsilon^ TA\epsilon)\), \(x(\epsilon,A)=E(\epsilon^ TA\epsilon -E\epsilon^ TA\epsilon)^ 4,\) \(\kappa_{\epsilon}=\min \{k:\) \(v(\epsilon\),A)\(\leq k tr((A\Sigma)^ 2)\), \(A\in {\mathfrak A}\},\) \(\lambda_{\epsilon}=\min \{\ell: x(\epsilon,A)\leq \ell v^ 2(\epsilon,A)\), \(A\in {\mathfrak A}\}.\) In the case when \(\epsilon =(\epsilon_ 1,...,\epsilon_ p)\) with independent \(\epsilon_ i\), \(i=1,...,p\), the upper bounds for \(v(\epsilon\),A) and \(x(\epsilon,A)\) are obtained, which give that \(\epsilon^ TA\epsilon -E\epsilon^ TA\epsilon \to 0\) a.s. as n (the dimension of \(\epsilon)\) increases and there exist constants \(\kappa \geq 2\), \(\lambda \geq 1\), \(c>0\) and \(\delta>{1\over2}\) such that \(\kappa_{\epsilon_ i}\leq \kappa\), \(\lambda \epsilon_ i\leq \lambda (i=1,...,p)\) and \(tr((A\Sigma)^ 2)\leq cn^{-\delta}\). Reviewer: T.Shervashidze Cited in 3 Documents MSC: 60E15 Inequalities; stochastic orderings 60F15 Strong limit theorems Keywords:upper bounds PDFBibTeX XMLCite \textit{J. Kleffe} and \textit{R. Thrum}, Math. Operationsforsch. Stat., Ser. Stat. 14, 211--216 (1983; Zbl 0545.60027) Full Text: DOI