Hoeffding's inequality

In probability theory describes the Hoeffding 's inequality ( Hoeffding by Vasily ) the maximum probability that a sum of independent random variables and limited deviates more than a constant from its expected value.

The Hoeffding 's inequality is also called the additive Chernoff inequality and is a special case of Bernstein 's inequality.

Set

Be independent random variables, so that almost surely. Further, let a positive, real-valued constant. Then:

Evidence

This proof follows the representation of D. Pollard, see also Lutz Dümbgens script ( see literature).

Consider for simplicity of notation, the random variables and also for a first on any real numbers monotonically increasing figure. After the Chebyshev 's inequality then:

Is due to the convexity of the exponential function

And it follows that

And for the constants. If we consider the logarithm of the right-hand side of this term

It can be shown by curve sketching and Taylor series expansion that always applies. If this value is due to the monotonicity of the exponential function as an upper bound in the first inequality again, we obtain

Resulting in an election of the assertion to be proved.

Examples

• Consider the following question: How likely is it to achieve a total of at least 500 eyes at hundertmaligem cubes? Describes the result of the dice roll with, so it follows with the Hoeffding 's inequality: