In probability theory, the Chernoff bound, named after Herman Chernoff but due to Herman Rubin, gives exponentially decreasing bounds on tail distributions of sums of independent random variables. It is a sharper bound than the known first or second moment based tail bounds such as Markov's inequality or Chebyshev inequality, which only yield power-law bounds on tail decay. However, the Chernoff bound requires that the variates be independent – a condition that neither the Markov nor the Chebyshev inequalities require.
It is related to the (historically prior) Bernstein inequalities, and to Hoeffding's inequality.
The generic Chernoff bound for a random variable X is attained by applying Markov's inequality to etX. For every :
When X is the sum of n random variables X1, ..., Xn, we get for any t > 0,
In particular, optimizing over t and using the assumption that Xi are independent, we obtain,