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Strong Laws for Urn Models with Balanced Replacement Matrices

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Strong Laws for Urn Models with Balanced Replacement Matrices
2.	Creator	Author's name, affiliation, country	Amites Dasgupta; Indian Statistical Institute; India
2.	Creator	Author's name, affiliation, country	Krishanu Maulik; Indian Statistical Institute; India
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Urn model, balanced triangular replacement matrix, Perron-Frobenius eigenvalue, irreducible matrix.
3.	Subject	Subject classification	Primary 60F15, 60F25, 60G42.
4.	Description	Abstract	We consider an urn model, whose replacement matrix has all entries nonnegative and is balanced, that is, has constant row sums. We obtain the rates of the counts of balls corresponding to each color for the strong laws to hold. The analysis requires a rearrangement of the colors in two steps. We first reduce the replacement matrix to a block upper triangular one, where the diagonal blocks are either irreducible or the scalar zero. The scalings for the color counts are then given inductively depending on the Perron-Frobenius eigenvalues of the irreducible diagonal blocks. In the second step of the rearrangement, the colors are further rearranged to reduce the block upper triangular replacement matrix to a canonical form. Under a further mild technical condition, we obtain the scalings and also identify the limits. We show that the limiting random variables corresponding to the counts of colors within a block are constant multiples of each other. We provide an easy-to-understand explicit formula for them as well. The model considered here contains the urn models with irreducible replacement matrix, as well as, the upper triangular one and several specific block upper triangular ones considered earlier in the literature and gives an exhaustive picture of the color counts in the general case with only possible restrictions that the replacement matrix is balanced and has nonnegative entries.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2011-09-19
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/928
10.	Identifier	Digital Object Identifier	10.1214/EJP.v16-928
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 16
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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