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Inequalities for moments of quadratic forms with applications to a.s. convergence. (English) Zbl 0545.60027

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

MSC:

60E15 Inequalities; stochastic orderings
60F15 Strong limit theorems

Keywords:

upper bounds
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