论文标题
改善了Hoeffding的引理和Hoeffding的尾巴界限
Improved Hoeffding's Lemma and Hoeffding's Tail Bounds
论文作者
论文摘要
这封信的目的是改善Hoeffding的引理，从而改善Hoeffding的尾巴范围。改进与左倾斜零平均随机变量$ x \ in [a，b] $，其中$ a <0 $和$ -a> b $。 Hoeffding改进的引理证据使用了Taylor的扩展，$ \ exp（sx），s \ in {\ bf r} $中的s \以及自1963年Hoeffding出版以来的$ -a> b $以来的最大数量（1--τ）的最大证明是hoeff $ -a 在端点而不是$τ= 0.5 $的情况下，例如$ b> -a $。使用Hoeffding的改进的引理，我们获得了单面和两个方面的尾巴界限，分别为$ P（s_n \ ge t）$和$ p（| s_n | \ ge t）$，其中 $ s_n = \ sum_ {i = 1}^nx_i $和[a_i，b_i]中的$ x_i \，i = 1，...，n $是独立的零平均随机变量（不一定相同分布）。有趣的是，我们还可以改善Hoeffding的两个方面绑定的所有$ \ {x_i：a_i \ ne b_i，i = 1，...，n \} $。之所以如此，是因为这里的单面绑定应增加$ p（-s_n \ ge t）$，其中左倾斜间隔变成右偏斜，反之亦然。
The purpose of this letter is to improve Hoeffding's lemma and consequently Hoeffding's tail bounds. The improvement pertains to left skewed zero mean random variables $X\in[a,b]$, where $a<0$ and $-a>b$. The proof of Hoeffding's improved lemma uses Taylor's expansion, the convexity of $\exp(sx), s\in {\bf R}$ and an unnoticed observation since Hoeffding's publication in 1963 that for $-a>b$ the maximum of the intermediate function $τ(1-τ)$ appearing in Hoeffding's proof is attained at an endpoint rather than at $τ=0.5$ as in the case $b>-a$. Using Hoeffding's improved lemma we obtain one sided and two sided tail bounds for $P(S_n\ge t)$ and $P(|S_n|\ge t)$, respectively, where $S_n=\sum_{i=1}^nX_i$ and the $X_i\in[a_i,b_i],i=1,...,n$ are independent zero mean random variables (not necessarily identically distributed). It is interesting to note that we could also improve Hoeffding's two sided bound for all $\{X_i: a_i\ne b_i,i=1,...,n\}$. This is so because here the one sided bound should be increased by $P(-S_n\ge t)$, wherein the left skewed intervals become right skewed and vice versa.
