Chernoff Bound - The First Step in The Proof of Chernoff Bounds

The First Step in The Proof of Chernoff Bounds

The Chernoff bound for a random variable X, which is the sum of n independent random variables, is obtained by applying etX for some well-chosen value of t. This method was first applied by Sergei Bernstein to prove the related Bernstein inequalities.

From Markov's inequality and using independence we can derive the following useful inequality:

For any t > 0,

In particular optimizing over t and using independence we obtain,

(1)

Similarly,

and so,

Read more about this topic:  Chernoff Bound

Famous quotes containing the words step, proof and/or bounds:

    Life begins at six—at least in the minds of six-year-olds. . . . In kindergarten you are the baby. In first grade you put down the baby. . . . Every first grader knows in some osmotic way that this is real life. . . . First grade is the first step on the way to a place in the grown-up world.
    Stella Chess (20th century)

    There are some persons in this world, who, unable to give better proof of being wise, take a strange delight in showing what they think they have sagaciously read in mankind by uncharitable suspicions of them.
    Herman Melville (1819–1891)

    How far men go for the material of their houses! The inhabitants of the most civilized cities, in all ages, send into far, primitive forests, beyond the bounds of their civilization, where the moose and bear and savage dwell, for their pine boards for ordinary use. And, on the other hand, the savage soon receives from cities iron arrow-points, hatchets, and guns, to point his savageness with.
    Henry David Thoreau (1817–1862)