## Concentration inequalities and martingale inequalities – a survey

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Venue: | Internet Math |

Citations: | 40 - 1 self |

### BibTeX

@ARTICLE{Chung_concentrationinequalities,

author = {Fan Chung and Linyuan Lu},

title = {Concentration inequalities and martingale inequalities – a survey},

journal = {Internet Math},

year = {}

}

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### Abstract

Abstract. We examine a number of generalized and extended versions of concentration inequalities and martingale inequalities. These inequalities are effective for analyzing processes with quite general conditions as illustrated in an example for an infinite Polya process and web graphs. 1.

### Citations

1707 | The Probabilistic Method
- Alon, Spencer
- 2000
(Show Context)
Citation Context ...robability spaces and if the incremental changes of the value of the event is “small”, then the martingale inequalities provide very good error estimates. The reader is referred to numerous textbooks =-=[5, 17, 20]-=- on this subject. In the past few years, there has been a great deal of research in analyzing general random graph models for realistic massive graphs which have uneven degree distribution such as the... |

1658 |
An introduction to probability theory and its
- Feller
- 1968
(Show Context)
Citation Context ... of the binomial distribution, after scaling, is the standard normal distribution and can be viewed as a special case of the Central-Limit Theorem, sometimes called the DeMoivre-Laplace limit Theorem =-=[15]-=-. 3sProbability 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 4600 4800 5000 5200 5400 value Figure 1: The Binomial distribution B(10000, 0.5) Probability density 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.... |

1392 | Statistical mechanics of complex networks
- Barabsi, Albert
- 2002
(Show Context)
Citation Context ...bject. In the past few years, there has been a great deal of research in analyzing general random graph models for realistic massive graphs which have uneven degree distribution such as the power law =-=[1, 2, 3, 4, 6]-=-. The usual concentration inequalities and martingale inequalities have often been found to be inadequate and in many cases not feasible. The reasons are multi-fold: Due to uneven degree distribution,... |

348 | A random graph model for massive graphs
- Aiello, Chung, et al.
- 2000
(Show Context)
Citation Context ...bject. In the past few years, there has been a great deal of research in analyzing general random graph models for realistic massive graphs which have uneven degree distribution such as the power law =-=[1, 2, 3, 4, 6]-=-. The usual concentration inequalities and martingale inequalities have often been found to be inadequate and in many cases not feasible. The reasons are multi-fold: Due to uneven degree distribution,... |

305 |
Emergence of scaling in random networks, Science 286
- Barabasi, Albert
- 1999
(Show Context)
Citation Context ...bject. In the past few years, there has been a great deal of research in analyzing general random graph models for realistic massive graphs which have uneven degree distribution such as the power law =-=[1, 2, 3, 4, 6]-=-. The usual concentration inequalities and martingale inequalities have often been found to be inadequate and in many cases not feasible. The reasons are multi-fold: Due to uneven degree distribution,... |

278 | A brief history of generative models for power law and lognormal distributions, in
- Mitzenmacher
(Show Context)
Citation Context ...ion of the preferential attachment scheme can be viewed as a variation of the Polya process as we will see. Before we proceed, we first give a short description for the preferential attachment scheme =-=[3, 21]-=-: • With probability p, forsomefixedp, add a new vertex v, andaddanedge {u, v} from v by randomly and independently choosing u in proportion to the degree of u in the current graph. The initial graph,... |

200 | The average distance in a random graph with given expected degrees
- Chung, lu
- 2003
(Show Context)
Citation Context ...ce all the work is quite recent and the selection is heavily influenced by our personal learning experience on this topic. Indeed, many of these inequalities have been included in our previous papers =-=[9,10,11,12]-=-. In addition to numerous variations of the inequalities, we also include an example of an application on a generalization of Polya’s urn problem. Due to the fundamental nature of these concentration ... |

160 |
Concentration. In Probabilistic methods for algorithmic discrete mathematics
- McDiarmid
(Show Context)
Citation Context ...robability spaces and if the incremental changes of the value of the event is “small”, then the martingale inequalities provide very good error estimates. The reader is referred to numerous textbooks =-=[5, 17, 20]-=- on this subject. In the past few years, there has been a great deal of research in analyzing general random graph models for realistic massive graphs which have uneven degree distribution such as the... |

135 | Connected components in random graphs with given degree sequences
- Chung, Lu
(Show Context)
Citation Context ...ce all the work is quite recent and the selection is heavily influenced by our personal learning experience on this topic. Indeed, many of these inequalities have been included in our previous papers =-=[9,10,11,12]-=-. In addition to numerous variations of the inequalities, we also include an example of an application on a generalization of Polya’s urn problem. Due to the fundamental nature of these concentration ... |

106 | Spectra of random graphs with given expected degrees
- Chung
- 2003
(Show Context)
Citation Context ...ce all the work is quite recent and the selection is heavily influenced by our personal learning experience on this topic. Indeed, many of these inequalities have been included in our previous papers =-=[9,10,11,12]-=-. In addition to numerous variations of the inequalities, we also include an example of an application on a generalization of Polya’s urn problem. Due to the fundamental nature of these concentration ... |

94 | A functional approach to external graph algorithms
- Abello, Buchsbaum, et al.
- 2002
(Show Context)
Citation Context |

88 |
Urn models and their application: an approach to modern discrete probability theory
- Johnson
- 1977
(Show Context)
Citation Context ... the ball in that bin; otherwise, place the ball in an existing bin, such that the probability the ball is placed in a bin is proportional to the number of balls in that bin. Polya’s urn problem (see =-=[18]-=-) is a special case of the above process with p = 0 so new bins are never created. For the case of p>0, this infinite Polya 33 λ 2sprocess has a similar flavor as the preferential attachment scheme, o... |

54 |
Concentration of multivariate polynomials and its applications
- Kim, Vu
(Show Context)
Citation Context ...ns are allowed to fail for relatively small subsets of the probability space and we can still have similar but weaker concentration inequalities. Similar techniques have been introduced by Kim and Vu =-=[19]-=- in their important work on deriving concentration inequalities for multivariate polynomials. The basic setup for decision trees can be found in [5] and has been used in the work of Alon, Kim and Spen... |

30 | Nearly perfect matchings in regular simple hypergraphs
- Alon, Kim, et al.
- 1997
(Show Context)
Citation Context ...their important work on deriving concentration inequalities for multivariate polynomials. The basic setup for decision trees can be found in [5] and has been used in the work of Alon, Kim and Spencer =-=[7]-=-. Wormald [22] considers martingales with a ‘stopping time’ that has a similar flavor. Here we use a rather general setting and we shall give a complete proof here. We are only interested in finite pr... |

27 | Coupling Online and Offline Analyses for Random Power Law Graphs
- Chung, Lu
(Show Context)
Citation Context |

25 |
A note on an inequality involving the normal distribution
- Chernoff
- 1981
(Show Context)
Citation Context ...ariables is to the expected value, various concentration inequalities are in play. 5sA typical version of the Chernoff inequalities, attributed to Herman Chernoff, can be stated as follows: Theorem 3 =-=[8]-=- Let X1,...,Xn be independent random variables with E(Xi) = 0 and |Xi| ≤1for all i. Let X = �n i=1 Xi and let σ2 bethevarianceofXi. Then Pr(|X| ≥kσ) ≤ 2e −k2 /4n , for any 0 ≤ k ≤ 2σ. If the random va... |

13 | 2003) “Generalizations of Polya’s urn problem - Chung, Handjani, et al. - 2010 |

3 |
Random evolution in massive graphs, Extended abstract appeared in: The 42th Annual Symposium on Foundation
- Aiello, Chung, et al.
- 2002
(Show Context)
Citation Context |

1 |
The differential equation method for random processes and greedy algorithms
- Wormald
- 1999
(Show Context)
Citation Context ...nt work on deriving concentration inequalities for multivariate polynomials. The basic setup for decision trees can be found in [5] and has been used in the work of Alon, Kim and Spencer [7]. Wormald =-=[22]-=- considers martingales with a ‘stopping time’ that has a similar flavor. Here we use a rather general setting and we shall give a complete proof here. We are only interested in finite probability spac... |