基于切尔诺夫界和中心极限定理的随机变量和的尾部概率的估计

Calculation of Tail Probability Based on Central Limit Theorem and Chernoff Bound

首先研究了独立同分布场合的中心极限定理以及切尔诺夫界,然后由这两个方法来估计n个随机变量和的尾部概率。在两种常见概率分布下随机变量和的尾部概率估算结果表明,利用中心极限定理,当随机变量个数n足够大时,可得到较准确的值;采用切尔诺夫界,虽然可以求出尾部概率的上界,但与真实值之间存在一定的误差。因此,若要用切尔诺夫不等式得到一个更加紧凑的界,须对切尔诺夫界作必要的修正。

Firstly, the central limit theorem for i. i. d （ independent and identically distributed） random variables and Chernoff bound are reviewed, with which the tail probability of n i. i.d. random variables is calculated. Results of the tail probability obtained under two common probability distributions suggest that central limit theorem can provide relatively precise values on condition that n is sufficiently large, while there always exist certain errors between the upper bound of the tail probability acquired by Chernoff bound and the exact value. Consequently, in order to obtain a tighter bound for the tail probability with Chernoff inequality, it＇s necessary to modify the Chernoff bound.