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

W P(White) = 3/8

B P(White) = 4/7

Intricate probabilities (depending on paths)

A = 1 2

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)

W BW

X = 1 persists

W 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 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?

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

Superlinear:

linear:

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 delayedand 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.