A Bound on Tail Probabilities for Quadratic Forms in Independent Random Variables Whose Distributions are not Necessarily Symmetric

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.