CIMPA Summer School 2014 Random structures, analytic and probabilistic approaches University An...
Transcript of CIMPA Summer School 2014 Random structures, analytic and probabilistic approaches University An...
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 ???