Order-of-magnitude bounds for expectations involving quadratic forms
Let \(X = (X_ 1, \dots, X_ n)\) be \(n\) independent r.v.s, \((a_{ij}) \in \mathbb{R}^{n \times n}\) a symmetric matrix with \(a_{jj} = 0\), and \(\Phi\) be a positive symmetric function s.t. \(\Phi (0) = 0\). If \(\Phi (2x) \leq 2^ \beta \Phi (x)\) holds and if \(\Phi (x)\) is convex and \(\Phi (\sqrt x)\) concave, the authors obtain upper and lower bounds for \(\mathbb{E} (\Phi (\sum^ n_{ij} a_{ij} X_ i X_ j))\) and \(\mathbb{E} (\max_{m \leq n} \Phi (\sum^ m_{ij} a_{ij} X_ i X_ j))\) giving the order of magnitude of the quadratic form. This result can be generalized to the case where for a \(k \in \mathbb{N}_ 0\) both \(\Phi (x^{2 - k})\) is convex and \(\Phi (x^{2 - k - 1})\) is concave. The proofs rely on a decoupling technique stating that for arbitrary convex functions \(\Psi\) the expressions \(\mathbb{E} (\Psi (\sum^ n_{ij} a_{ij} X_ i X_ j))\) and \(\mathbb{E} (\max_{m \leq n} \Psi (\sum^ m_{ij} a_{ij} X_ i X_ j))\) can be compared from above and below with \(\mathbb{E} (\Psi (\sum^ n_{ij} a_{ij} X_ i \widetilde X_ j))\) and \(\mathbb{E} (\max_{m \leq n} \Psi (\sum^ m_{ij} a_{ij} X_ i \widetilde X_ j))\), respectively. Here, \(\widetilde X\) denotes an independent copy of \(X\). This technique also yields a version of Khinchin's inequality, namely \[ \mathbb{E} \Bigl( \sum^ n_{ij} a_{ij} X_ i X_ j \Bigr) \sim \mathbb{E} \biggl( \Bigl( \sum^ n_{ij} a^ 2_{ij} X^ 2_ iX^ 2_ j \Bigr)^{1/2} \biggr) \sim \mathbb{E} \biggl( \Bigl( \sum^ n_{ij} a^ 2_{ij} X ^ 2_ i \widetilde X^ 2_ j \Bigr)^{1/2} \biggl). \]
scientific article; zbMATH DE number 681393