To install click the Add extension button. That's it.
The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.
How to transfigure the Wikipedia
Would you like Wikipedia to always look as professional and up-to-date? We have created a browser extension. It will enhance any encyclopedic page you visit with the magic of the WIKI 2 technology.
Try it — you can delete it anytime.
Install in 5 seconds
Yep, but later
4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let X1, ..., Xn be independent Bernoulli random variables taking values +1 and −1 with probability 1/2 (this distribution is also known as the Rademacher distribution), then for every positive ,
Bernstein inequalities were proven and published by Sergei Bernstein in the 1920s and 1930s.[1][2][3][4] Later, these inequalities were rediscovered several times in various forms. 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]
YouTube Encyclopedic
1/5
Views:
1 191
743
47 023
19 507
22 645
The Bernstein Sato polynomial: Bernstein's inequality
1. Let be independent zero-mean random variables. Suppose that almost surely, for all Then, for all positive ,
2. Let be independent zero-mean random variables. Suppose that for some positive real and every integer ,
Then
3. Let be independent zero-mean random variables. Suppose that
for all integer Denote
Then,
4. Bernstein also proved generalizations of the inequalities above to weakly dependent random variables. For example, inequality (2) can be extended as follows. Let be possibly non-independent random variables. Suppose that for all integers ,
Then
More general results for martingales can be found in Fan et al. (2015).[7]
Proofs
The proofs are based on an application of Markov's inequality to the random variable
for a suitable choice of the parameter .
Generalizations
The Bernstein inequality can be generalized to Gaussian random matrices. Let be a scalar where is a complex Hermitian matrix and is complex vector of size . The vector is a Gaussian vector of size . Then for any , we have
where is the vectorization operation and where is the largest eigenvalue of . The proof is detailed here.[8] Another similar inequality is formulated as
^S.N.Bernstein, "On a modification of Chebyshev's inequality and of the error formula of Laplace" vol. 4, #5 (original publication: Ann. Sci. Inst. Sav. Ukraine, Sect. Math. 1, 1924)
^Bernstein, S. N. (1937). "Об определенных модификациях неравенства Чебышева" [On certain modifications of Chebyshev's inequality]. Doklady Akademii Nauk SSSR. 17 (6): 275–277.
^S.N.Bernstein, "Theory of Probability" (Russian), Moscow, 1927
^J.V.Uspensky, "Introduction to Mathematical Probability", McGraw-Hill Book Company, 1937
^Freedman, D.A. (1975). "On tail probabilities for martingales". Ann. Probab. 3: 100–118.