The class of tenable zero- balanced Pólya urn schemes: characterization and Gaussian phases
Hosam M. Mahmoud
The George Washington University
Joint work with Sanaa kholfi
Talk at 22 Workshop on Analysis of Algorithms
June 13, 2011
Plan
Pólya urn schemes Explain the title: Irreducible nondegenerate zero-balanced
tenable schemes (the class Ck) Characterization The Markov chain Gaussian phases Examples
Pólya Urns
We have a starting urn of white and blue ballsWe have rules of evolution (ball addition matrix)
A = a b
c d
W
B
BW
pick
add
General setup
Can be positive or negative (application in trees) The entries are generally random
Who said analysis of algorithms cannot be colorful?
References
Norman Johnson and Samuel Kotz (1977).
Urn Models and Their Applications.
Wily, New York, USA. Hosam Mahmoud (2008).
Pólya Urn Models.
Chapman-Hall, Florida, USA.
Zero-Balance
a b
c d
When a+b = c+d =K
we say the urn is balanced,
and the balance factor is K
we look at urns with balance 0
Reducible and Irreducible schemes
The urn associated with coupon collection
is reducible. If we start monochromatically with
blue balls, white balls never appear.
Degeneracy
Suppose we have a k-color scheme in which a color (say pink) is always an exact multiple of another (say crimson), say twice as many (initially, and the rules keep this proportion).
We can combine the two colors into one (say the simply très chic red), study a scheme of dimensionality k-1, then restore the structure of each color.
Characterization
Can the matrix
3 -6 3 0 2 -2
0 0 0 0 0 0
-1 0 1 0 0 0
0 0 0 0 -1 1
0 0 0 0 1 -1
be in our class?
Examples
Ehrenfest
3x3 deterministic
3x3 random
B and B’ are independent
Bernoulli (½) random variables
Necessity
These conditions are also necessary. If we assume the conditions, the replacement matrix must be in the form given.
Phases
Sublinear: log log n, log n, n¼, n½, n¾, n/log n
Linear: 5n, (7 + 2 (-1)n) n
Superlinear: n log n, n2, en
Dominant color
three colors: initially, n – 2└log n┘, └ log n┘, └log n┘
X0(n) = α n + o (n) = 1
0 n + + o (n) 0 Remark: In the presence of dominant colors in critical cases different scale factors are
needed for different colors and there may or may not be a single multivariate central limit theorem for all the colors .
Conditional variance
Deterministic approximations in each phase:
Sublinear:
Linear and superlinear: mean
An example with an initially dominant color
When the row corresponding to the
dominant color is deterministic