The class of tenable zerobalanced 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 balls
We have rules of evolution (ball addition matrix)
add
W B
pick
A
=
W
a b
B
c d
Intricate probabilities (depending on paths)
P(White) = 3/8
W
B
P(White) = 4/7
1 2
A =
2 0
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.
Tenable urns
As an example of what we have in mind
Where X = Bernoulli(p)
X = 1 persists
W
W
W
B
W
?
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
Ehrenfest
A model for the mixing of gases
Reducible and Irreducible
schemes
The urn associated with coupon collection
is reducible. If we start monochromatically with
blue balls, white balls never appear.
Communication problems
between colors
This urn is like is two noncommunicating
Ehrenfest schemes
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 0 0
-1 0 1
0 0 0
0 0 0
be in our class?
0
0
0
0
0
2
0
0
-1
1
-2
0
0
1
-1
Characterization
Examples
Ehrenfest
3x3 deterministic
3x3 random
B and B’ are independent
Bernoulli (½) random variables
Sufficient conditions
Proof.
Necessity
These conditions are also necessary. If we assume
the conditions, the replacement matrix must be in
the form given.
Markov chain
v = (v1, …, vk) is a left eigenvector of
E[A].
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
Theorem 3 (main result)
In the absence of a dominant color
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 .
Stochastic recurrence
Let 1j,r (n) be the indicator of picking color r at the jth step
The mean
where
Martingales
E[Xj | Fj-1] = Xj-1
Fair gambling:
E[Xj | Fj-1] = Xj-1 + (-1) x ½ + (+1) x ½
The underlying matingale
Recall that
Centered martingale
Martingale conditions
Uniformly bounded differences
Matrix norm calculation
Conditional Lindeberg’s
condition
Conditional variance
Deterministic approximations in each phase:
Sublinear:
Linear and superlinear: mean
Examples
Ehrenfest
Sublinear:
Ehrenfest in its linear and
superlinear phases
linear:
Superlinear:
A 3-color scheme with random
replacements
Sublinear
superlinear
An example with a degenerate
scheme
Reduced nondegenerate
scheme
An example with an initially
dominant color
When the row corresponding to the
dominant color is deterministic
The sublinear phase is delayed
and the scales are different
When the row has random
entries
Conclusion




Analysis in phases reveals subtle dynamics and
rates of convergence
The class is a good candidate for “analytic urn
methodology”
The methodology may carry over to other
reducible urns
The methodology may carry over to other
balanced urns.