# Reference request: binomial tail is greater than Gaussian tail

I remember there is a result from the early 1980s, which states that the tail probability of a binomial distribution is always at least as large as the tail probability of the normal distribution (at least when $p \leq 1/4$ or something like this for the binomial distribution is assumed). However, I cannot rememeber the name of the theorem and the author, resp. Can someone help me?

• The comparison is rather vice versa: the normal tail is heavier than the binomial, at least far enough from the mean, where the binomial tail is just 0. – Iosif Pinelis Feb 19 '18 at 12:05
• Oh well, probably there is a requirement to stay away from the extreme tail. Still, there is a theorem which says that (within some range, probably) the tail of the binomial distribution is heavier than that of the normal distribution. I am sure it exists, I just cannot find it anymore. – Kurisuto Asutora Feb 19 '18 at 12:43
• you should find the answer in this earlier MO question – Carlo Beenakker Feb 19 '18 at 18:32
• No, that's not what I mean. As Iosif pointed out the desired result cannot hold for the whole tail, but only up to some reasonable threshold. However, the range for the Berry-Essen is too small as a large deviations inequality, for the purpose that I have in mind. – Kurisuto Asutora Feb 20 '18 at 6:57

• It is amazing to me that the condition $p\le1/4$ is enough for the binomial tail $P(B_{n,p}\ge k)$ to be heavier than the corresponding normal tail for all integers $k$ between $np$ and $n$. Of course, here one cannot replace $\ge k$ by $>k$. – Iosif Pinelis Feb 21 '18 at 3:33