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 the first, step, proof and/or bounds:
“Show me a man who has acted, and who has not been the victim and slave of his action. What they have done commits and enforces them to do the same again. The first act, which was to be an experiment, becomes a sacrament.”
—Ralph Waldo Emerson (1803–1882)
“It is not when truth is dirty, but when it is shallow, that the lover of knowledge is reluctant to step into its waters.”
—Friedrich Nietzsche (1844–1900)
“If we view our children as stupid, naughty, disturbed, or guilty of their misdeeds, they will learn to behold themselves as foolish, faulty, or shameful specimens of humanity. They will regard us as judges from whom they wish to hide, and they will interpret everything we say as further proof of their unworthiness. If we view them as innocent, or at least merely ignorant, they will gain understanding from their experiences, and they will continue to regard us as wise partners.”
—Polly Berrien Berends (20th century)
“Prohibition will work great injury to the cause of temperance. It is a species of intemperance within itself, for it goes beyond the bounds of reason in that it attempts to control a man’s appetite by legislation, and makes a crime out of things that are not crimes. A Prohibition law strikes a blow at the very principles upon which our government was founded.”
—Abraham Lincoln (1809–1865)