CIMPA Summer School 2014

Random structures, analytic and probabilistic approaches

University An Najah, Nablus (Palestine),Nicolas Pouyanne

P

´

olya urn models

– slides –

First steps: histories (1)

On the x-axis, the number of red balls in the urn after n drawings.

On the y-axis, the number of histories of length n.

R = I2 =

✓1 00 1

◆, U0 =

✓25

◆

n = 1 n = 2

Red balls in the original Polya urn I2 after n drawings

First steps: histories (2)

On the x-axis, the number of red balls in the urn after n drawings.

On the y-axis, the number of histories of length n.

R = I2 =

✓1 00 1

◆, U0 =

✓25

◆

n = 1 n = 2 n = 3 n = 10

Red balls in the original Polya urn I2 after n drawings

First steps: histories (3)

On the x-axis, the number of red balls in the urn after n drawings.

On the y-axis, the number of histories of length n.

R = R1 :=

✓1 1211 2

◆, U0 =

✓10

◆

n = 1 n = 2 n = 3 n = 10

Red balls in the small Polya urn R1 after n drawings

First steps: histories (4)

On the x-axis, the number of red balls in the urn after n drawings.

On the y-axis, the number of histories of length n.

R = R2 :=

✓12 12 11

◆, U0 =

✓10

◆

n = 1 n = 2 n = 3 n = 10

Red balls in the large Polya urn R2 after n drawings

First steps: histories (5)

n = 1 n = 2 n = 3 n = 10Red balls in the original Polya urn I2 after n drawings, initial composition (2, 5)

n = 1 n = 2 n = 3 n = 10Red balls in the small urn R1 after n drawings, initial composition (1, 0)

n = 1 n = 2 n = 3 n = 10Red balls in the large urn R2 after n drawings, initial composition (1, 0)

First steps: trajectories (1)

On the x-axis, the number of drawings up to N ;

On the y-axis, the number U(1)n

of red balls in the urn.

R = I2 =

✓1 00 1

◆, U0 =

✓25

◆

N = 100 N = 1000 N = 50000

Red balls in three sequences of N drawings in an original Polya urn I2

First steps: trajectories (2)

On the x-axis, the number of drawings up to N ;

On the y-axis, the number U(1)n

of red balls in the urn.

R = R1 :=

✓1 1211 2

◆, U0 =

✓10

◆

N = 100 N = 1000 N = 50000

Red balls in three sequences of N drawings in an small urn R1

First steps: trajectories (3)

On the x-axis, the number of drawings up to N ;

On the y-axis, the number U(1)n

of red balls in the urn.

R = R2 :=

✓12 12 11

◆, U0 =

✓10

◆

N = 100 N = 1000 N = 50000

Red balls in three sequences of N drawings in an large urn R2

First steps: trajectories (4)

N = 100 N = 1000 N = 50000Red balls in three sequences of N drawings in an original Polya urn I2

N = 100 N = 1000 N = 50000Red balls in three sequences of N drawings in a small urn R1

N = 100 N = 1000 N = 50000Red balls in three sequences of N drawings in a large urn R2

Original Polya urn: trajectories (1)

Theorem 1 (Polya original urn)

Suppose that the urn is Polya’s original one, i.e. that R = I2. Then, as n

tends to infinity,

U

n

Sn

�!n!1

D

almost surely and in any Lp

, p � 1, where D is a Dirichlet distributed

2-dimensional random vector with parameter

✓↵

S

,

�

S

◆.

Normalized number of red balls 1n

U

(1)n

in three sequences of 100 drawings

in an original Polya urn I2,

initial composition (2, 5).

Original Polya urn: trajectories (2)

On the x-axis, the number of drawings up to N ;

On the y-axis, the normalised number of red balls 1n

U

(1)n

.

R = I2 =

✓1 00 1

◆, U0 =

✓25

◆

N = 100 N = 1000 N = 50000

Red balls in three sequences of N drawings in an original Polya urn I2

Original Polya urn: asymptotic distribution

R = I2 =

✓1 00 1

◆, U0 =

✓25

◆

Interpolated distribution of the normalised

number of red balls in a Polya urn I2

after n=200 drawings.

On the x-axis: 1n

⇣U

(1)n

� EU

(1)n

⌘.

On the y-axis, probability.

Density of a centered

Beta (2, 5) distribution.

Small urn: trajectories (1)

Theorem 2 (Small urns)

Suppose that the urn is small, which means that � < 1/2. Then as n

tends to infinity,

(i)

U

n

n

converges to v1, almost surely and in any Lp

, p � 1;

(ii) assume further that R is not triangular. Then,

U

n

� nv1pn

converges in

distribution to a centered gaussian vector [+formula covariance].

Normalised number of red balls 1n

U

(1)n

in three sequences of 100 drawings

in an small urn R1 =

✓1 1211 2

◆,

initial composition (1, 0).

Small urn: trajectories (2)

On the x-axis, the number of drawings up to N ;

On the y-axis, the normalised number of red balls 1n

U

(1)n

.

R = R1 =

✓1 1211 2

◆, U0 =

✓10

◆

N = 100 N = 1000 N = 50000

Red balls in three sequences of N drawings in a small urn R1

Small urn: trajectories (3)

Theorem 3 (Small urns)

Suppose that the urn is small, which means that � < 1/2. Then as n

tends to infinity,

(i)

U

n

n

converges to v1, almost surely and in any Lp

, p � 1;

(ii) assume further that R is not triangular. Then,

U

n

� nv1pn

converges in

distribution to a centered gaussian vector [+formula covariance].

Completely normalised number of red

balls 1pn

⇣U

(1)n

� EU

(1)n

⌘in three

sequences of 100 drawings in an small urn

R1 =

✓1 1211 2

◆, initial composition (1, 0).

Small urn: trajectories (4)

On the x-axis, the number of drawings up to N ;

On the y-axis, the completely normalised number of red balls 1pn

⇣U

(1)n

� EU

(1)n

⌘.

R = R1 =

✓1 1211 2

◆, U0 =

✓10

◆

N = 100 N = 1000 N = 50000

Red balls in three sequences of N drawings in a small urn R1

Small urn: asymptotic distribution

R = R1 =

✓1 1211 2

◆, U0 =

✓10

◆

Distribution of the normalised number

of red balls in a small urn R1 after

n=600 drawings.

On the x-axis: 1pn

⇣U

(1)n

� EU

(1)n

⌘.

On the y-axis, probability.

Density of a centered normal

distribution with variance1

1�2�bcm

2

(b+c)2= 5200

529 .

Large urn: trajectories (1)

Theorem 4 (Large urns)

Suppose that the urn is large, which means that 1/2 < � < 1. Then as

n tends to infinity,

(i)

U

n

n

converges to v1, almost surely and in any Lp

, p � 1;

(ii)

U

n

� nv1

n

�

converges almost surely and in any Lp

, p � 1 to Wv2 where

W is a real-valued random variable [+formula expectation].

Normalised number of red balls 1n

U

(1)n

in three sequences of 100 drawings

in an large urn R2 =

✓12 12 11

◆,

initial composition (1, 0).

Large urn: trajectories (2)

On the x-axis, the number of drawings up to N ;

On the y-axis, the normalised number of red balls 1n

U

(1)n

.

R = R2 =

✓12 12 11

◆, U0 =

✓10

◆

N = 100 N = 1000 N = 50000

Red balls in three sequences of N drawings in a small urn R1

Large urn: trajectories (3)

Theorem 5 (Large urns)

Suppose that the urn is large, which means that 1/2 < � < 1. Then as

n tends to infinity,

(i)

U

n

n

converges to v1, almost surely and in any Lp

, p � 1;

(ii)

U

n

� nv1

n

�

converges almost surely and in any Lp

, p � 1 to Wv2 where

W is a real-valued random variable [+formula expectation].

Completely normalised number of red

balls1

n

�

⇣U

(1)n

� EU

(1)n

⌘in three

sequences of 100 drawings in an large urn

R2 =

✓12 12 11

◆, initial composition (1, 0).

Large urn: trajectories (4)

On the x-axis, the number of drawings up to N ;

On the y-axis, the completely normalised number of red balls1

n

�

⇣U

(1)n

� EU

(1)n

⌘.

R = R2 =

✓12 12 11

◆, U0 =

✓10

◆

N = 100 N = 1000 N = 50000

Red balls in three sequences of N drawings in a small urn R1

Large urn: asymptotic distribution (1)

R = R2 =

✓12 12 11

◆, U0 =

✓10

◆

Distribution of the normalised number

of red balls in a large urn R1 after

n=800 drawings.

On the x-axis:1

n

�

⇣U

(1)n

� EU

(1)n

⌘.

On the y-axis, probability.

?

B. Chauvin et N. Pouyanne, UVSQ 2014, LSMA523 1/3

What is this law ?

Large urn: asymptotic distribution (2)

The distribution W of a large urn depends on the initial composition.

R = R2 =

✓12 12 11

◆

(↵, �) = (1, 0) (↵, �) = (1, 1) (↵, �) = (2, 1)

Normalised distribution of the number of red balls in a large urn R2

after 500 drawings, initial composition (↵, �)

Polya, small or large: a brief resume

U

(1)n

1n

U

(1)n

1

n

⇤

⇣U

(1)n

� nv

(1)1

⌘Asympt. distribution

Polya

- - -

B. Chauvin et N. Pouyanne, UVSQ 2014, LSMA523 1/3

Beta

Small Gauss

Large ???