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In [[probability theory]], the '''Azuma–Hoeffding inequality''' (named after [[Kazuoki Azuma]] and [[Wassily Hoeffding]]) gives a [[concentration inequality|concentration result]] for the values of [[martingale (probability theory)|martingale]]s that have bounded differences.
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Suppose { ''X''<sub>''k''</sub> : ''k'' = 0, 1, 2, 3, ... } is a [[Martingale (probability theory)|martingale]] (or [[Martingale_(probability_theory)#Submartingales.2C_supermartingales.2C_and_relationship_to_harmonic_functions|super-martingale]]) and
 
:<math>|X_k - X_{k-1}| < c_k, \, </math>
 
[[almost surely]]. Then for all positive integers ''N'' and all positive [[real number|reals]] ''t'',
 
:<math>P(X_N - X_0 \geq t) \leq \exp\left ({-t^2 \over 2 \sum_{k=1}^N c_k^2} \right). </math>
 
And symmetrically (when ''X''<sub>''k''</sub> is a sub-martingale):
 
:<math>P(X_N - X_0 \leq -t) \leq \exp\left ({-t^2 \over 2 \sum_{k=1}^N c_k^2} \right). </math>
 
If ''X'' is a martingale, using both inequalities above and applying the [[union bound]] allows one to obtain a two-sided bound:
 
:<math>P(|X_N - X_0| \geq t) \leq 2\exp\left ({-t^2 \over 2 \sum_{k=1}^N c_k^2} \right). </math>
 
Azuma's inequality applied to the [[Doob martingale]] gives the [[method of bounded differences]] (MOBD) which is common in the analysis of [[randomized algorithm]]s.
 
==Simple example of Azuma's inequality for coin flips==
Let ''F''<sub>''i''</sub> be a sequence of independent and identically distributed random coin flips (i.e., let ''F''<sub>''i''</sub> be equally likely to be -1 or 1 independent of the other values of ''F''<sub>''i''</sub>). Defining <math>X_i = \sum_{j=1}^i F_j</math> yields a [[Martingale (probability theory)|martingale]] with |''X''<sub>''k''</sub>&nbsp;&minus;&nbsp;''X''<sub>''k''&minus;1</sub>|&nbsp;≤&nbsp;1, allowing us to apply Azuma's inequality.  Specifically, we get
 
:<math> \Pr[X_N > t] \leq \exp\left(\frac{-t^2}{2 N}\right).</math>
 
For example, if we set ''t'' proportional to ''N'', then this tells us that although the ''maximum'' possible value of ''X''<sub>''N''</sub> scales linearly with ''N'', the ''probability'' that the sum scales linearly with ''N'' [[exponential decay|decreases exponentially fast]] with&nbsp;''N''.
 
==Remark==
 
A [[Bernstein inequalities (probability theory)|similar inequality]] was proved under weaker assumptions by [[Sergei Bernstein]] in 1937.
 
Hoeffding proved this result for independent variables rather than martingale differences, and also observed that slight modifications of his argument establish the result for martingale differences (see page 18 of his 1963 paper).
 
==See also==
* [[McDiarmid's inequality]]
* [[Markov inequality]]
* [[Hoeffding's inequality]]
* [[Chebyshev's inequality]]
* [[Bernstein inequalities (probability theory)]]
* [[Bennett's inequality]]
 
{{No footnotes|date=July 2010}}
 
==References==
 
* {{cite book|first1=N. |last1=Alon |first2= J. |last2=Spencer|title=The Probabilistic Method|publisher= Wiley|location=New York|year= 1992}}
* {{cite journal|doi=10.2748/tmj/1178243286|first=K. |last=Azuma|title=Weighted Sums of Certain Dependent Random Variables|journal=[[Tôhoku Mathematical Journal]]|volume=19|pages= 357–367 |year=1967|issue=3|url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.tmj/1178243286|format=PDF}}
* {{cite journal| last=Bernstein | first=Sergei N. | authorlink=Sergei Natanovich Bernstein | year=1937 |trans_title=On certain modifications of Chebyshev's inequality |language=Russian | journal=Doklady Akademii Nauk SSSR | volume=17 | issue=6 | pages=275–277 |title=На определенных модификациях неравенства Чебишева}} (vol. 4, item 22 in the collected works)
* {{cite book| first=C. |last=McDiarmid|chapter= On the method of bounded differences|title=Surveys in Combinatorics|series= London Math. Soc. Lectures Notes 141|publisher= Cambridge Univ. Press|location= Cambridge |year=1989|pages=148–188}}
* {{Cite journal|doi=10.2307/2282952|first1=W. |last1=Hoeffding|title=Probability inequalities for sums of bounded random variables|journal=Journal of the American Statistical Association|volume=58|issue=301|pages= 13–30|year= 1963|mr=  0144363  }}
* {{Cite journal|first1=A. P. |last1=Godbole |first2= P. |last2=Hitczenko|title=Beyond the method of bounded differences|journal=DIMACS Series in Discrete Mathematics and Theoretical Computer Science|volume= 41|pages=43–58|year= 1998 |mr=1630408  }}
 
[[Category:Probabilistic inequalities]]
[[Category:Martingale theory]]

Latest revision as of 13:28, 6 November 2014

At any time, you should remember you have a fantastic lover in winter season-UGG boots. I was requested these kinds of a dilemma by my pal "Which organ do you assume is the most important for the entire human body? Is it face, neck, hand, pores and skin or foot?" I answered her without the need of hesitation. "Foot." Because all the acupuncture details of the human human body are on the foot. Wintertime is coming. It has come to be a puzzling subject matter on how to safeguard your feet. Low-cost UGG is an substitute heat boot.

UGG boots are manufactured from Australian parts of small sheepskin or picked calfskin with Australian wool. Calf leather and Australian wool are incredibly high-priced products. They are lined with complete central and seem in assorted hues on the outside. Normal substantial good quality merino wool pores and skin for toes totally free pursuits and respiration produced a extremely cozy space, as on the delicate the clouds dancing, you are cost-free to go your individual way. Affordable UGG can preserve your feet heat in cold winter devoid of donning any socks. The reason is the exceptional internal fibres circulate heat air which can warm cold ft extremely immediately, and warm them in temperatures as lower as minus thirty degrees celsius.

It is not denied that UGGs sale channels are far more and a lot more flexible. We can pick out UGG retailers or counters which provide all types of series and vibrant types. In addition to, UGG on the net is also a excellent option. The rate is decrease than stores'. You can absolutely manage these boots.

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Some individuals are unable to assist inquiring that which sort of shoes could go well with their apparel. According to Brian Fong in his short article "What to use with UGG boots?", "There are no trend principles when it will come to wearing these (UGGs) boots." So whatever model you belong to, from punk to prep, you can use the UGGs any way you want.

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