Extending Hoeffding's Inequality for Supermatingales

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Mirshani, Ardalan | 2013

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 45660 (02)
University: Sharif University of Technology
Department: Mathematical Science
Advisor(s): Z.Zangeneh, Bijan
Abstract:
In this thesis, we are going to extend hoeffding's inequality to supermartingale with differences bounded from above and compare this inequality with the other ones which derived in the past. After defining basic thorems, we introduce upper bounds which are derived for the probability that the sum of some independent random variables exceeds its mean by a positive number and also, compare to each o ther. In addition , we use these bounds and derive upper bounds of probability such as above for dependent random variables. Then, we introduce a n inequality which cover Hoeffding's inequality for independent random variables and also extend it to the case of supermartingales with differences bounded from above. Finally, when we compare this inequality to the other well-known ones, we conclude that in some situations, this is better than the other inequalities
Keywords:
Martingales ; Upper Bound Method ; Hoeffding's Inequality ; Supermartingale