Abstract
Let $\{X_i\}^\infty_{i = -\infty}$ be a sequence of independent random variables with zero means and let $P\lbrack |X_i| \geqq x \rbrack \leqq M \int^\infty_x \exp\{-\Upsilon t^2\} dt$ for all $x \geqq 0$ where $M$ and $\Upsilon$ are positive constants. Let $((a_{ij}))^\infty_{i, j = -\infty}$ be an infinite matrix of real numbers with $a_{ij} = a_{ji}$ for all $i, j$ and $\Lambda^2 = \sum_{i,j}a^2_{ij} < \infty$. Let $\|A\|$ be the norm of $A = ((|a_{ij}|))$ considered as an operator on $l_2$ and set $S = \sum_{i,j}a_{ij}(X_iX_j - E(X_i X_j))$. In this note it is shown that there exist positive constants $C_1$ and $C_2$ depending only on $M$ and $\Upsilon$ such that $P\lbrack S \geqq \varepsilon \rbrack \leqq \exp \{-\min (C_{1 \varepsilon}/\|A\|, C_{2\varepsilon^2}/\Lambda^2)\}$ for all $\varepsilon > 0$. This result has previously been established in the literature for sequences of random variables which have symmetric distributions.
Citation
F. T. Wright. "A Bound on Tail Probabilities for Quadratic Forms in Independent Random Variables Whose Distributions are not Necessarily Symmetric." Ann. Probab. 1 (6) 1068 - 1070, December, 1973. https://doi.org/10.1214/aop/1176996815
Information