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Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality and McDiarmid's inequality. It is similar to the Chernoff bound, but tends to be less sharp, in particular when the variance of the random variables is small. [2] It is similar to, but incomparable with, one of Bernstein's inequalities.
Chernoff bounds may also be applied to general sums of independent, bounded random variables, regardless of their distribution; this is known as Hoeffding's inequality. The proof follows a similar approach to the other Chernoff bounds, but applying Hoeffding's lemma to bound the moment generating functions (see Hoeffding's inequality).
Thus, special cases of the Bernstein inequalities are also known as the Chernoff bound, Hoeffding's inequality and Azuma's inequality. The martingale case of the Bernstein inequality is known as Freedman's inequality [5] and its refinement is known as Hoeffding's inequality. [6]
Placing addition assumption that the summands in Matrix Azuma are independent gives a matrix extension of Hoeffding's inequalities. Consider a finite sequence { X k } {\displaystyle \{\mathbf {X} _{k}\}} of independent, random, self-adjoint matrices with dimension d {\displaystyle d} , and let { A k } {\displaystyle \{\mathbf {A} _{k}\}} be a ...
Such inequalities are of importance in several fields, including communication complexity (e.g., in proofs of the gap Hamming problem [13]) and graph theory. [14] An interesting anti-concentration inequality for weighted sums of independent Rademacher random variables can be obtained using the Paley–Zygmund and the Khintchine inequalities. [15]
This is not news to anyone who has followed reports of growing wealth inequality in America. It's still rather shocking. Imagine it -- a mere tenth of the top 1% of U.S. households has more than ...
Hoeffding's inequality is the Chernoff bound obtained using this fact. Convolutions. Density of a mixture of three normal distributions μ ...
Poison Profits. A HuffPost / WNYC investigation into lead contamination in New York City