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Randomness of Restricted Value Martingales, Selection Rules, and Graph Sequences.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	Randomness of Restricted Value Martingales, Selection Rules, and Graph Sequences./
Author:	Pardo, John J.
Description:	1 online resource (65 pages)
Notes:	Source: Dissertation Abstracts International, Volume: 79-04(E), Section: B.
Contained By:	Dissertation Abstracts International79-04B(E).
Subject:	Mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9780355331387

Randomness of Restricted Value Martingales, Selection Rules, and Graph Sequences.
Pardo, John J.

Randomness of Restricted Value Martingales, Selection Rules, and Graph Sequences. - 1 online resource (65 pages)

Source: Dissertation Abstracts International, Volume: 79-04(E), Section: B.

Thesis (Ph.D.)

Includes bibliographical references

In this dissertation we consider the classical notions of stochasticity and randomness as they pertain to several areas of mathematics. We look at three main topics: martingales, selection rules, and graphs.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2018

Mode of access: World Wide Web

ISBN: 9780355331387Subjects--Topical Terms:

527692
Mathematics.
Index Terms--Genre/Form:

554714
Electronic books.

Randomness of Restricted Value Martingales, Selection Rules, and Graph Sequences.
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100 1 $a Pardo, John J. $3 1180978
245 1 0 $a Randomness of Restricted Value Martingales, Selection Rules, and Graph Sequences.
264 0 $c 2017
300 $a 1 online resource (65 pages)
336 $a text $b txt $2 rdacontent
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338 $a online resource $b cr $2 rdacarrier
500 $a Source: Dissertation Abstracts International, Volume: 79-04(E), Section: B.
500 $a Adviser: Jan S. Reimann.
502 $a Thesis (Ph.D.) $c The Pennsylvania State University $d 2017.
504 $a Includes bibliographical references
520 $a In this dissertation we consider the classical notions of stochasticity and randomness as they pertain to several areas of mathematics. We look at three main topics: martingales, selection rules, and graphs.
520 $a First, we discuss the notion of martingales and focus our attention onto restricted value martingales; this allows us to make some insights into the notion of A-valued randomness for a set A of real numbers. Martingales are the classical mathematical representation of a betting strategy; they are defined in a way that nicely accommodates the notion of "win your bet if you are successful and lose your bet if you are unsuccessful." By looking at restricted value martingales, we are able to focus on betting strategies that only allow wagers of certain amounts. Bavly and Peretz defined notions of anticipation and evasion to relate the relative strengths of restricted value martingales; this provides a natural way to compare A-valued randomness and B-valued randomness for different sets A, B &sube; R +. Essentially, A evades B when there is an A-martingale that can succeed against any countable set of B-martingales, and otherwise B anticipates A -- to get the similar notion of single evasion one only needs to consider a single B-martingale. Bavly and Peretz characterized the evasion/anticipation relationships when A is bounded and B is bounded away from 0, as well as when B is well-ordered, however the question of what happens when B has 0 as an accumulation point remained open. We look at various geometric sequences for our set B to help answer this question with regard to single evasion/single anticipation.
520 $a Second, we consider selection rules and how they relate to normality and randomness. Traditionally, a selection rule indicates how to select a subsequence from a given binary sequence based on the initial segment of the sequence that has already been viewed. Clearly one must ask what sort of properties are preserved under the application of a selection rule, however there are two equally important ways to formulate this question: for a fixed set of sequences what selection rules always output an element of that set when applied to an element of that set, and for a fixed set of selection rules what properties of sequences are preserved? Von Mises made the intuitions of these questions rigorous with the notions of kollektivs and admissible selection rules. We focus on sequences and selection rules defined in terms of computability, normality, Martin-Lof randomness, and the strong law of large number.
520 $a Lastly, we apply selection rules to graphs and graph sequences as well as discuss several notions of randomness for graphs. Graphs as mathematical objects are intuitively more structurally complicated than binary sequences, however both graphs and sequences are useful to consider in the context of information content. As a result, it makes sense to look at any probabilistic notion that applies to binary sequences -- stochasticity, normality, randomness -- as it applies to graph objects. We define selection rules for graphs and show that homogeneous countably infinite graphs are preserved under selection by a certain class of selection rules. We also consider several possible characterizations of randomness for graphs.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2018
538 $a Mode of access: World Wide Web
650 4 $a Mathematics. $3 527692
655 7 $a Electronic books. $2 local $3 554714
690 $a 0405
710 2 $a ProQuest Information and Learning Co. $3 1178819
710 2 $a The Pennsylvania State University. $b Mathematics. $3 1180974
773 0 $t Dissertation Abstracts International $g 79-04B(E).
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10666596 $z click for full text (PQDT)