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Fluctuations of martingales and winning probabilities of game contestants

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Fluctuations of martingales and winning probabilities of game contestants
2.	Creator	Author's name, affiliation, country	David Aldous; University of California, Berkeley; United States
2.	Creator	Author's name, affiliation, country	Mykhaylo Shkolnikov; University of California, Berkeley; United States
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	entrance boundary, fluctuations, martingale; upcrossing; Wright-Fisher diffusion
3.	Subject	Subject classification	60G44; 91A60
4.	Description	Abstract	Within a contest there is some probability M_i(t) that contestant  i will be the winner, given information available at time t, and M_i(t) must be a martingale in t.  Assume continuous paths, to capture the idea that relevant information is acquired slowly. Provided each contestant's initial winning probability is at most b, one can easily calculate, without needing further model specification, the expectations of the random variables  N_b = number of contestants whose winning probability ever exceeds b, and D_{ab} = total number of downcrossings of the martingales over an interval [a,b]. The distributions of N_b and D_{ab} do depend on further model details, and we study how concentrated or spread out the distributions can be. The extremal models for N_b correspond to two contrasting intuitively natural methods for determining a winner: progressively shorten a list of remaining candidates, or sequentially examine candidates to be declared winner or eliminated. We give less precise bounds on the variability of D_{ab}. We formalize the setting of infinitely many contestants each with infinitesimally small chance of winning, in which the explicit results are more elegant. A canonical process in this setting is the  Wright-Fisher diffusion associated with an infinite population of initially distinct alleles; we show how this process fits our setting and  raise the problem of finding the distributions of N_b and D_{ab} for this process.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	NSF
7.	Date	(YYYY-MM-DD)	2013-04-08
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/2422
10.	Identifier	Digital Object Identifier	10.1214/EJP.v18-2422
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 18
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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