P olya urns via the contraction method Ralph Neininger Institute - - PowerPoint PPT Presentation

SLIDE 1

P´

• lya urns via the contraction method

Ralph Neininger Institute for Mathematics J.W. Goethe-University Frankfurt a.M. AofA 2013 Menorca, Spain joint work with M. Knape arXiv:1301.3404

SLIDE 2

P´

• lya urns

Initial configuration: r0 red balls, b0 blue balls. Replacement matrix: red blue red a b blue c d a, d ∈ N0 ∪ {−1}, b, c ∈ N0

SLIDE 3

Methods

Main focus: # balls of each color, n → ∞ Calculations with moment generating function Calculations with moments Martingale methods Embedding into continuous time branching processes Counting urn histories + analytic combinatorics Here: Recursive approach

a + b = c + d: “balanced urn”

SLIDE 4

Methods

Main focus: # balls of each color, n → ∞ Calculations with moment generating function Calculations with moments Martingale methods Embedding into continuous time branching processes Counting urn histories + analytic combinatorics Here: Recursive approach

a + b = c + d: “balanced urn”

SLIDE 5

Methods

Main focus: # balls of each color, n → ∞ Calculations with moment generating function Calculations with moments Martingale methods Embedding into continuous time branching processes Counting urn histories + analytic combinatorics Here: Recursive approach

a + b = c + d: “balanced urn”

SLIDE 6

A discrete-time embedding

red blue red 1 4 blue 3 2

SLIDE 7

A discrete-time embedding

red blue red 1 4 blue 3 2

SLIDE 8

A discrete-time embedding

red blue red 1 4 blue 3 2

SLIDE 9

A discrete-time embedding

red blue red 1 4 blue 3 2

SLIDE 10

A discrete-time embedding

red blue red 1 4 blue 3 2

SLIDE 11

A discrete-time embedding

red blue red 1 4 blue 3 2

SLIDE 12

A discrete-time embedding

red blue red 1 4 blue 3 2

SLIDE 13

A discrete-time embedding

red blue red 1 4 blue 3 2

SLIDE 14

A discrete-time embedding

red blue red 1 4 blue 3 2

SLIDE 15

A discrete-time embedding

red blue red 1 4 blue 3 2

SLIDE 16

A discrete-time embedding

red blue red 1 4 blue 3 2

SLIDE 17

Embedding into trees

Balls are held in leaves of the tree. Balanced urn: a+b = c+d =: K−1. Branch degree: K. Recurrence: # red balls =

K

• j=1

# red balls in j-th subtree. Subtrees rooted by red / blue (ghost) balls behave differently.

SLIDE 18

Embedding into trees

Balls are held in leaves of the tree. Balanced urn: a+b = c+d =: K−1. Branch degree: K. Recurrence: # red balls =

K

• j=1

# red balls in j-th subtree. Subtrees rooted by red / blue (ghost) balls behave differently.

SLIDE 19

Embedding into trees

Balls are held in leaves of the tree. Balanced urn: a+b = c+d =: K−1. Branch degree: K. Recurrence: # red balls =

K

• j=1

# red balls in j-th subtree. Subtrees rooted by red / blue (ghost) balls behave differently.

SLIDE 20

Embedding into trees

Balls are held in leaves of the tree. Balanced urn: a+b = c+d =: K−1. Branch degree: K. Recurrence: # red balls =

K

• j=1

# red balls in j-th subtree. Subtrees rooted by red / blue (ghost) balls behave differently.

SLIDE 21

General recurrence

R(r)

n : # red balls when starting with red after n steps.

R(b)

n : # red balls when starting with blue after n steps.

I(n) = (I1, . . . , IK): vector of # draws in each subtree. a + b = c + d = K − 1 balanced urn. R(r)

n d

=

a+1

• j=1

R(r), j

Ij

+

K

• j=a+2

R(b), j

Ij

R(b)

n d

=

c

• j=1

R(r), j

Ij

+

K

• j=c+1

R(b), j

Ij

SLIDE 22

General recurrence

R(r)

n : # red balls when starting with red after n steps.

R(b)

n : # red balls when starting with blue after n steps.

I(n) = (I1, . . . , IK): vector of # draws in each subtree. a + b = c + d = K − 1 balanced urn. R(r)

n d

=

a+1

• j=1

R(r), j

Ij

+

K

• j=a+2

R(b), j

Ij

R(b)

n d

=

c

• j=1

R(r), j

Ij

+

K

• j=c+1

R(b), j

Ij

SLIDE 23

General recurrence

R(r)

n : # red balls when starting with red after n steps.

R(b)

n : # red balls when starting with blue after n steps.

I(n) = (I1, . . . , IK): vector of # draws in each subtree. a + b = c + d = K − 1 balanced urn. R(r)

n d

=

a+1

• j=1

R(r), j

Ij

+

K

• j=a+2

R(b), j

Ij

R(b)

n d

=

c

• j=1

R(r), j

Ij

+

K

• j=c+1

R(b), j

Ij

SLIDE 24

General recurrence

R(r)

n : # red balls when starting with red after n steps.

R(b)

n : # red balls when starting with blue after n steps.

I(n) = (I1, . . . , IK): vector of # draws in each subtree. a + b = c + d = K − 1 balanced urn. R(r)

n d

=

a+1

• j=1

R(r), j

Ij

+

K

• j=a+2

R(b), j

Ij

R(b)

n d

=

c

• j=1

R(r), j

Ij

+

K

• j=c+1

R(b), j

Ij

SLIDE 25

General recurrence

R(r)

n : # red balls when starting with red after n steps.

R(b)

n : # red balls when starting with blue after n steps.

I(n) = (I1, . . . , IK): vector of # draws in each subtree. a + b = c + d = K − 1 balanced urn. R(r)

n d

=

a+1

• j=1

R(r), j

Ij

+

K

• j=a+2

R(b), j

Ij

R(b)

n d

=

c

• j=1

R(r), j

Ij

+

K

• j=c+1

R(b), j

Ij

SLIDE 26

Sizes of subtrees

I(n) = (I1, . . . , IK): vector of # draws in each subtree. Label balls by their subtree. P´

• lya urn for the labels.

Initial condition: (1, 1, . . . , 1) Replacement matrix: diag(K − 1, K − 1, . . . , K − 1) 1 nIn

a.s.

− → (D1, . . . , DK) With (D1, . . . , DK) ∼ Dirichlet

• 1

K − 1, . . . , 1 K − 1

• .

SLIDE 27

Sizes of subtrees

I(n) = (I1, . . . , IK): vector of # draws in each subtree.

1 1 1 1 1 1 2 3 3 3 3 3 3 3 3 3 3 3 4 5 6

Label balls by their subtree. P´

• lya urn for the labels.

Initial condition: (1, 1, . . . , 1) Replacement matrix: diag(K − 1, K − 1, . . . , K − 1) 1 nI(n) a.s. − → (D1, . . . , DK) With (D1, . . . , DK) ∼ Dirichlet

• 1

K − 1, . . . , 1 K − 1

• .

SLIDE 28

Sizes of subtrees

I(n) = (I1, . . . , IK): vector of # draws in each subtree.

1 1 1 1 1 1 2 3 3 3 3 3 3 3 3 3 3 3 4 5 6

Label balls by their subtree. P´

• lya urn for the labels.

Initial condition: (1, 1, . . . , 1) Replacement matrix: diag(K − 1, K − 1, . . . , K − 1) 1 nI(n) a.s. − → (D1, . . . , DK) With (D1, . . . , DK) ∼ Dirichlet

• 1

K − 1, . . . , 1 K − 1

• .

SLIDE 29

Sizes of subtrees

I(n) = (I1, . . . , IK): vector of # draws in each subtree.

1 1 1 1 1 1 2 3 3 3 3 3 3 3 3 3 3 3 4 5 6

Label balls by their subtree. P´

• lya urn for the labels.

Initial condition: (1, 1, . . . , 1) Replacement matrix: diag(K − 1, K − 1, . . . , K − 1) 1 nI(n) a.s. − → (D1, . . . , DK) With (D1, . . . , DK) ∼ Dirichlet

• 1

K − 1, . . . , 1 K − 1

• .

SLIDE 30

Sizes of subtrees

I(n) = (I1, . . . , IK): vector of # draws in each subtree.

1 1 1 1 1 1 2 3 3 3 3 3 3 3 3 3 3 3 4 5 6

Label balls by their subtree. P´

• lya urn for the labels.

Initial condition: (1, 1, . . . , 1) Replacement matrix: diag(K − 1, K − 1, . . . , K − 1) 1 nI(n) a.s. − → (D1, . . . , DK) With (D1, . . . , DK) ∼ Dirichlet

• 1

K − 1, . . . , 1 K − 1

• .

SLIDE 31

Normalization

R(r)

n d

=

a+1

• j=1

R(r), j

Ij

+

K

• j=a+2

R(b), j

Ij

R(b)

n d

=

c

• j=1

R(r), j

Ij

+

K

• j=c+1

R(b), j

Ij

Normalization X(r)

n

:= R(r)

n

− E R(r)

n

nγL(n) , X(b)

n

:= R(b)

n

− E R(b)

n

nγL(n) Modified equations X(r)

n d

=

a+1

• j=1

Ij

n

γ L(Ij)

L(n) X(r), j

Ij

+

K

• j=a+2

Ij

n

γ L(Ij)

L(n) X(b), j

Ij

X(b)

n d

= similarly

SLIDE 32

Normalization

R(r)

n d

=

a+1

• j=1

R(r), j

Ij

+

K

• j=a+2

R(b), j

Ij

R(b)

n d

=

c

• j=1

R(r), j

Ij

+

K

• j=c+1

R(b), j

Ij

Normalization X(r)

n

:= R(r)

n

− E R(r)

n

nγL(n) , X(b)

n

:= R(b)

n

− E R(b)

n

nγL(n) Modified equations X(r)

n d

=

a+1

• j=1

Ij

n

γ L(Ij)

L(n) X(r), j

Ij

+

K

• j=a+2

Ij

n

γ L(Ij)

L(n) X(b), j

Ij

X(b)

n d

= similarly

SLIDE 33

Normalization

R(r)

n d

=

a+1

• j=1

R(r), j

Ij

+

K

• j=a+2

R(b), j

Ij

R(b)

n d

=

c

• j=1

R(r), j

Ij

+

K

• j=c+1

R(b), j

Ij

Normalization X(r)

n

:= R(r)

n

− E R(r)

n

nγL(n) , X(b)

n

:= R(b)

n

− E R(b)

n

nγL(n) Modified equations X(r)

n d

=

a+1

• j=1

Ij

n

γ L(Ij)

L(n) X(r), j

Ij

+

K

• j=a+2

Ij

n

γ L(Ij)

L(n) X(b), j

Ij

+ T (n)

r

X(b)

n d

= similarly

SLIDE 34

Fixed-point equations

X(r)

n d

=

a+1

• j=1

Ij

n

γ L(Ij)

L(n) X(r), j

Ij

+

K

• j=a+2

Ij

n

γ L(Ij)

L(n) X(b), j

Ij

+ T (n)

r

X(b)

n d

= similarly Limit X(r)

d

=

a+1

• j=1

Dγ

j X(r), j + K

• j=a+2

Dγ

j X(b), j

X(b)

d

=

c

• j=1

Dγ

j X(r), j + K

• j=c+1

Dγ

j X(b), j

Calls for recursive methods: Contraction method

SLIDE 35

Fixed-point equations

X(r)

n d

=

a+1

• j=1

Ij

n

γ L(Ij)

L(n) X(r), j

Ij

+

K

• j=a+2

Ij

n

γ L(Ij)

L(n) X(b), j

Ij

+ T (n)

r

X(b)

n d

= similarly Limit X(r)

d

=

a+1

• j=1

Dγ

j X(r), j + K

• j=a+2

Dγ

j X(b), j + Tr

X(b)

d

=

c

• j=1

Dγ

j X(r), j + K

• j=c+1

Dγ

j X(b), j + Tb

Calls for recursive methods: Contraction method

SLIDE 36

Fixed-point equations

X(r)

n d

=

a+1

• j=1

Ij

n

γ L(Ij)

L(n) X(r), j

Ij

+

K

• j=a+2

Ij

n

γ L(Ij)

L(n) X(b), j

Ij

+ T (n)

r

X(b)

n d

= similarly Limit X(r)

d

=

a+1

• j=1

Dγ

j X(r), j + K

• j=a+2

Dγ

j X(b), j + Tr

X(b)

d

=

c

• j=1

Dγ

j X(r), j + K

• j=c+1

Dγ

j X(b), j + Tb

Calls for recursive methods: Contraction method

SLIDE 37

More colors

Types (colors) 1, . . . , m. R[j]

n : # type 1 balls, n steps, start with one type j ball.

E [R[j]

n ] =

          

cµn + djnλ + o(nλ) λ > 1/2 γ = λ cµn + o(√n) γ = 1/2 cµn + ℜ(κjniµ)nλ + o(nλ) λ > 1/2 γ = λ + iµ System of limits: X[j] d =

m

• i=1
• r∈Jij

(Dr)γX[i],r + b[j], j = 1, . . . , m.

SLIDE 38

More colors

Types (colors) 1, . . . , m. R[j]

n : # type 1 balls, n steps, start with one type j ball.

E [R[j]

n ] =

          

cµn + djnλ + o(nλ) λ > 1/2 γ = λ cµn + o(√n) γ = 1/2 cµn + ℜ(κjniµ)nλ + o(nλ) λ > 1/2 γ = λ + iµ System of limits: X[j] d =

m

• i=1
• r∈Jij

(Dr)γX[i],r + b[j], j = 1, . . . , m.

SLIDE 39

More colors

Types (colors) 1, . . . , m. R[j]

n : # type 1 balls, n steps, start with one type j ball.

E [R[j]

n ] =

          

cµn + djnλ + o(nλ) λ > 1/2 γ = λ cµn + o(√n) γ = 1/2 cµn + ℜ(κjniµ)nλ + o(nλ) λ > 1/2 γ = λ + iµ System of limits: X[j] d =

m

• i=1
• r∈Jij

(Dr)γX[i],r + b[j], j = 1, . . . , m.

SLIDE 40

Limit map

X(r)

d

=

a+1

• j=1

Dγ

j X(r), j + K

• j=a+2

Dγ

j X(b), j + Tr

X(b)

d

=

c

• j=1

Dγ

j X(r), j + K

• j=c+1

Dγ

j X(b), j + Tb

Corresponding limit map: T : M × M → M × M (µ, ν) →

• L

a+1

• j=1

Dγ

j Wj + K

• j=a+2

Dγ

j Zj + Tr

• ,

L

• c
• j=1

Dγ

j Wj + K

• j=c+1

Dγ

j Zj + Tb

• ,

where the Wj, Zj are indep enden t and L(Wj) = µ, L(Zj) = ν for all j.

SLIDE 41

Limit map

X(r)

d

=

a+1

• j=1

Dγ

j X(r), j + K

• j=a+2

Dγ

j X(b), j + Tr

X(b)

d

=

c

• j=1

Dγ

j X(r), j + K

• j=c+1

Dγ

j X(b), j + Tb

Corresponding limit map: T : M × M → M × M (µ, ν) →

• L

a+1

• j=1

Dγ

j Wj + K

• j=a+2

Dγ

j Zj + Tr

• ,

L

• c
• j=1

Dγ

j Wj + K

• j=c+1

Dγ

j Zj + Tb

• ,

where the Wj, Zj are indep enden t and L(Wj) = µ, L(Zj) = ν for all j.

SLIDE 42

Limit map

X(r)

d

=

a+1

• j=1

Dγ

j X(r), j + K

• j=a+2

Dγ

j X(b), j + Tr

X(b)

d

=

c

• j=1

Dγ

j X(r), j + K

• j=c+1

Dγ

j X(b), j + Tb

Corresponding limit map: T : M × M → M × M (µ, ν) →

• L

a+1

• j=1

Dγ

j Wj + K

• j=a+2

Dγ

j Zj + Tr

• ,

L

• c
• j=1

Dγ

j Wj + K

• j=c+1

Dγ

j Zj + Tb

• ,

where the Wj, Zj are independent and L(Wj) = µ, L(Zj) = ν for all j.

SLIDE 43

Metrics

Useful metrics on M: ℓp: minimal Lp-metric ζs: Zolotarev metric On appropriate subspaces of M × M: ℓ∨

p ((µ1, ν1), (µ2, ν2))

:= max{ℓp(µ1, µ2), ℓp(ν1, ν2)} ζ∨

s ((µ1, ν1), (µ2, ν2))

:= max{ζs(µ1, µ2), ζs(ν1, ν2)} E [R[j]

n ] =

          

cµn + djnλ + o(nλ) λ > 1/2 γ = λ cµn + o(√n) γ = 1/2 cµn + ℜ(κjniµ)nλ + o(nλ) λ > 1/2 γ = λ + iµ

SLIDE 44

Metrics

Useful metrics on M: ℓp: minimal Lp-metric ζs: Zolotarev metric On appropriate subspaces of M × M: ℓ∨

p ((µ1, ν1), (µ2, ν2))

:= max{ℓp(µ1, µ2), ℓp(ν1, ν2)} ζ∨

s ((µ1, ν1), (µ2, ν2))

:= max{ζs(µ1, µ2), ζs(ν1, ν2)} E [R[j]

n ] =

          

cµn + djnλ + o(nλ) λ > 1/2 γ = λ cµn + o(√n) γ = 1/2 cµn + ℜ(κjniµ)nλ + o(nλ) λ > 1/2 γ = λ + iµ

SLIDE 45

Another example

Random replacement:

 

Bα 1 − Bα 1 − Bβ Bβ

  ,

α, β ∈ [0, 1] Bα: Bernoulli(α) Bβ: Bernoulli(β) Smythe & Rosenberger (1995), Smythe (1996), Bai et al. (1999, 2002), Janson (2004).

SLIDE 46

Recurrences

R(r)

n : # red balls, initial red,

R(b)

n : # red balls, initial blue.

R(r)

n d

= R(r)

In + Bα

R(r)

n+1−In + (1 − Bα)

R(b)

n+1−In

R(b)

n d

= R(b)

In + Bβ

R(b)

n+1−In + (1 − Bβ)

R(r)

n+1−In

In uniform{1, . . . , n} distributed.

SLIDE 47

Recurrences

R(r)

n : # red balls, initial red,

R(b)

n : # red balls, initial blue.

R(r)

n d

= R(r)

In + Bα

R(r)

n+1−In + (1 − Bα)

R(b)

n+1−In

R(b)

n d

= R(b)

In + Bβ

R(b)

n+1−In + (1 − Bβ)

R(r)

n+1−In

In uniform{1, . . . , n} distributed.

SLIDE 48

Limit equations

Case α + β ≤ 3/2: X(r)

d

= √ UX(r) + √ 1 − UBα X(r) + √ 1 − U(1 − Bα)X(b), X(b)

d

= √ UX(b) + √ 1 − UBβ X(b) + √ 1 − U(1 − Bα)X(r). Case α + β > 3/2: Set γ := α + β − 1. X(r)

d

= UγX(r) + (1 − U)γBα X(r) + (1 − U)γ(1 − Bα)X(b) + Tr, X(b)

d

= UγX(b) + (1 − U)γBβ X(b) + (1 − U)γ(1 − Bα)X(r) + Tb. In both systems X(r), X(r), X(b), X(b), U independent.

SLIDE 49

Bivariate formulation

Rn :=

  R(r)

n

R(b)

n

  .

Then Rn

d

=

• 1

1

• RIn +
• Bα

1 − Bα 1 − Bβ Bβ

• Rn+1−In

(Rj)j, ( Rj)j, Bα, Bβ, In independent. Coupling: Urns starting with red resp. blue ball are coupled.

SLIDE 50

Bivariate formulation

Rn :=

  R(r)

n

R(b)

n

  .

Then Rn

d

=

• 1

1

• RIn +
• Bα

1 − Bα 1 − Bβ Bβ

• Rn+1−In

(Rj)j, ( Rj)j, Bα, Bβ, In independent. Coupling: Urns starting with red resp. blue ball are coupled.

SLIDE 51

Bivariate formulation

Rn :=

  R(r)

n

R(b)

n

  .

Then Rn

d

=

• 1

1

• RIn +
• Bα

1 − Bα 1 − Bβ Bβ

• Rn+1−In

(Rj)j, ( Rj)j, Bα, Bβ, In independent. Coupling: Urns starting with red resp. blue ball are coupled.

SLIDE 52

Limit equation

(I) α + β ≤ 3/2. Xn := 1 √n(Rn − E Rn) Limit equation:

• X1

X2

• d

= √ U

• X1

X2

• +

√ 1 − U

• Bα

1 − Bα 1 − Bβ Bβ X1

• X2
• Bivariate normal distribution solves.

Do not have contraction in the whole range!

SLIDE 53

Limit equation

(I) α + β ≤ 3/2. Xn := 1 √n(Rn − E Rn) Limit equation:

• X1

X2

• d

= √ U

• X1

X2

• +

√ 1 − U

• Bα

1 − Bα 1 − Bβ Bβ X1

• X2
• Bivariate normal distribution solves.

Do not have contraction in the whole range!

SLIDE 54

Limit equation

(I) α + β ≤ 3/2. Xn := 1 √n(Rn − E Rn) Limit equation:

• X1

X2

• d

= √ U

• X1

X2

• +

√ 1 − U

• Bα

1 − Bα 1 − Bβ Bβ X1

• X2
• Bivariate normal distribution solves.

Do not have contraction in the whole range!

SLIDE 55

Limit equation

(I) α + β ≤ 3/2. Xn := 1 √n(Rn − E Rn) Limit equation:

• X1

X2

• d

= √ U

• X1

X2

• +

√ 1 − U

• Bα

1 − Bα 1 − Bβ Bβ X1

• X2
• Bivariate normal distribution solves.

Do not have contraction in the whole range!

SLIDE 56

Contraction condition

E [U3/2] + E [(1 − U)3/2]E

 

• Bα

1 − Bα 1 − Bβ Bβ

• 3
• p

  < 1.

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

α β

α +β > 3/2 ξ <1 ξ >1 ξ >1

SLIDE 57

Contraction condition

Non-normal case α + β > 3/2:

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

α β

α +β < 3/2 ξ2 <1 ξ2 >1 ξ2 >1

SLIDE 58

Contraction condition

Non-normal case α + β > 3/2:

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

α β

α +β < 3/2 ξ3 <1 ξ3 >1 ξ3 >1

SLIDE 59

Systems versus multivariate

System formulation: X(r)

d

= √ UX(r) + √ 1 − UBα X(r) + √ 1 − U(1 − Bα)X(b), X(b)

d

= √ UX(b) + √ 1 − UBβ X(b) + √ 1 − U(1 − Bα)X(r). Work space: M(R) × M(R) (Precise: MR

3(0, 1) × MR 3(0, 1))

Bivariate formulation:

• X1

X2

• d

= √ U

• X1

X2

• +

√ 1 − U

• Bα

1 − Bα 1 − Bβ Bβ X1

• X2
• Work space: M(R × R)

(Precise: MR2

3 (0, Id2)).

SLIDE 60

Systems versus multivariate

System formulation: X(r)

d

= √ UX(r) + √ 1 − UBα X(r) + √ 1 − U(1 − Bα)X(b), X(b)

d

= √ UX(b) + √ 1 − UBβ X(b) + √ 1 − U(1 − Bα)X(r). Work space: M(R) × M(R) (Precise: MR

3(0, 1) × MR 3(0, 1))

Bivariate formulation:

• X1

X2

• d

= √ U

• X1

X2

• +

√ 1 − U

• Bα

1 − Bα 1 − Bβ Bβ X1

• X2
• Work space: M(R × R)

(Precise: MR2

3 (0, Id2)).

SLIDE 61

Systems versus multivariate

System formulation: X(r)

d

= √ UX(r) + √ 1 − UBα X(r) + √ 1 − U(1 − Bα)X(b), X(b)

d

= √ UX(b) + √ 1 − UBβ X(b) + √ 1 − U(1 − Bα)X(r). Work space: M(R) × M(R) (Precise: MR

3(0, 1) × MR 3(0, 1))

Bivariate formulation:

• X1

X2

• d

= √ U

• X1

X2

• +

√ 1 − U

• Bα

1 − Bα 1 − Bβ Bβ X1

• X2
• Work space: M(R × R)

(Precise: MR2

3 (0, Id2)).

SLIDE 62

Cyclic urns

m ≥ 2 types (colors) R =

       

1 1 ... 1 1

       

R[j]

n : # type 1 balls after n steps starting with one type j ball.

Janson (1983, 2004, 2006), Pouyanne (2005, 2008)

SLIDE 63

Recurrences

R[j]

n : # type 1 balls after n steps starting with one type j ball.

R[1]

n d

= R[1]

In + R[2] Jn ,

R[2]

n d

= R[2]

In + R[3] Jn ,

. . . R[m]

n d

= R[m]

In

+ R[1]

Jn ,

In: uniform on {0, . . . , n − 1}. E

• R[j]

n

• =

          

n m + o(√n),

2 ≤ m ≤ 5

n m + Θ(√n),

m = 6

n m + ℜ(κjniµ)nλ + o(nλ),

m ≥ 7.

SLIDE 64

Recurrences

R[j]

n : # type 1 balls after n steps starting with one type j ball.

R[1]

n d

= R[1]

In + R[2] Jn ,

R[2]

n d

= R[2]

In + R[3] Jn ,

. . . R[m]

n d

= R[m]

In

+ R[1]

Jn ,

In: uniform on {0, . . . , n − 1}. E

• R[j]

n

• =

          

n m + o(√n),

2 ≤ m ≤ 5

n m + Θ(√n),

m = 6

n m + ℜ(κjniµ)nλ + o(nλ),

m ≥ 7.

SLIDE 65

Limit equations

2 ≤ m ≤ 6: With U unif[0, 1]: X[1]

d

= √ UX[1] + √ 1 − UX[2], X[2]

d

= √ UX[2] + √ 1 − UX[3], . . . X[m]

d

= √ UX[m] + √ 1 − UX[1], m ≥ 7: With ω := e2πi/m: X[1]

d

= UωX[1] + (1 − U)ωX[2], X[2]

d

= UωX[2] + (1 − U)ωX[3], . . . X[m]

d

= UωX[m] + (1 − U)ωX[1].

SLIDE 66

Limit equations

2 ≤ m ≤ 6: With U unif[0, 1]: X[1]

d

= √ UX[1] + √ 1 − UX[2], X[2]

d

= √ UX[2] + √ 1 − UX[3], . . . X[m]

d

= √ UX[m] + √ 1 − UX[1], m ≥ 7: With ω := e2πi/m: X[1]

d

= UωX[1] + (1 − U)ωX[2], X[2]

d

= UωX[2] + (1 − U)ωX[3], . . . X[m]

d

= UωX[m] + (1 − U)ωX[1].

SLIDE 67

Periodic case m ≥ 7

System reduces: X d = UωX + ω(1 − U)ωX′ in MC

2

• 2

mΓ(ω + 1)

• ,

(cf. Janson ALEA06) Then X[j] d = ωj−1X, j = 1, . . . , m. Asymptotic periodic behavior: ℓ2

 R[j]

n − n m

nλ , ℜ

• e

i

• µ ln(n)+2πj−1

m

• Λ

  → 0

(n → ∞).

SLIDE 68

Periodic case m ≥ 7

System reduces: X d = UωX + ω(1 − U)ωX′ in MC

2

• 2

mΓ(ω + 1)

• ,

(cf. Janson ALEA06) Then X[j] d = ωj−1X, j = 1, . . . , m. Asymptotic periodic behavior: ℓ2

 R[j]

n − n m

nλ , ℜ

• e

i

• µ ln(n)+2πj−1

m

• Λ

  → 0

(n → ∞).

SLIDE 69

Periodic case m ≥ 7

System reduces: X d = UωX + ω(1 − U)ωX′ in MC

2

• 2

mΓ(ω + 1)

• ,

(cf. Janson ALEA06) Then X[j] d = ωj−1X, j = 1, . . . , m. Asymptotic periodic behavior: ℓ2

 R[j]

n − n m

nλ , ℜ

• e

i

• µ ln(n)+2πj−1

m

• X

  → 0

(n → ∞).