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Small deviations of sums of random v...

Garnett, Brian.

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Small deviations of sums of random variables.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Small deviations of sums of random variables./
Author:	Garnett, Brian.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2016,
Description:	59 p.
Notes:	Source: Dissertation Abstracts International, Volume: 78-04(E), Section: B.
Contained By:	Dissertation Abstracts International78-04B(E).
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10291807
ISBN:	9781369349894

Small deviations of sums of random variables.
Garnett, Brian.

Small deviations of sums of random variables. - Ann Arbor : ProQuest Dissertations & Theses, 2016 - 59 p.

Source: Dissertation Abstracts International, Volume: 78-04(E), Section: B.

Thesis (Ph.D.)--Rutgers The State University of New Jersey - New Brunswick, 2016.

In this thesis, we study the probability of a small deviation from the mean of a sum of independent or semi-independent random variables. In contrast with the rich history of large deviation inequalities, small deviations have only recently gained attention, and we make contributions to several problems on this topic.

ISBN: 9781369349894Subjects--Topical Terms:

515831
Mathematics.

Small deviations of sums of random variables.
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100 1 $a Garnett, Brian. $3 3343190
245 1 0 $a Small deviations of sums of random variables.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2016
300 $a 59 p.
500 $a Source: Dissertation Abstracts International, Volume: 78-04(E), Section: B.
500 $a Adviser: Swastik Kopparty.
502 $a Thesis (Ph.D.)--Rutgers The State University of New Jersey - New Brunswick, 2016.
520 $a In this thesis, we study the probability of a small deviation from the mean of a sum of independent or semi-independent random variables. In contrast with the rich history of large deviation inequalities, small deviations have only recently gained attention, and we make contributions to several problems on this topic.
520 $a Perhaps the most significant result in this field was an inequality proved by Feige. Let X1, . . . , Xn be nonnegative independent random variables, with E[ Xi] ≤ 1 ∀i, and let X = sum(n/i=1) X i. Then for any n, Pr[X 0, for some alpha ≥ 1/13. This bound was later improved to 1/8 by He, Zhang, and Zhang. Building off their work, we improve the bound to approximately .14. The conjectured true bound is 1/e &sime; .368, so there is still (possibly) quite a gap left to fill.
520 $a We also consider whether or not such small deviation inequalities hold for k-wise independent random variables. We show that for some classes of random variables, 4- wise independence is sufficient for a constant lower bound of alpha = 1/6, which we show to be tight. Furthermore, we present counterexamples showing that 3-wise independence is insufficient for a positive constant lower bound.
520 $a For sums of Bernoulli random variables, we can let alpha = 1/ e. We also show that k-wise independence can bring us arbitrarily close to that bound for large enough k.
590 $a School code: 0190.
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
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710 2 $a Rutgers The State University of New Jersey - New Brunswick. $b Graduate School - New Brunswick. $3 1019196
773 0 $t Dissertation Abstracts International $g 78-04B(E).
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792 $a 2016
793 $a English
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