Bootstrap random walks

(1)

Bootstrap random walks

Kais Hamza

Monash University

Joint work with Andrea Collevecchio & Meng Shi

(2)

Bootstrap random walks

Introduction

The two and three dimensional processes Higher Iterations

An extension – any (prime) number of values

Beyond the product (work in progress)

An application, a connection and more

(3)

Introduction

The two and three dimensional processes Higher Iterations

An extension – any (prime) number of values

Beyond the product (work in progress)

An application, a connection and more

(4)

Bootstrap random walks Introduction

The model setting

I Let (ξ n ) n be a sequence of independent identically distributed random variables such that

P(ξ i = −1) = P(ξ i = +1) = 1 2 .

I Let η _k = Q k

j =1 ξ _j . Then (η _n ) _n≥0 = (ξ ^d _n ) _n≥0 .

I The σ-algebras generated by these two sequences are the same, i.e.

σ(ξ 1 , · · · , ξ n ) = σ(η 1 , · · · , η n ).

I Let X n = P n

k=1 ξ k and Y n = P n

k=1 η k , with X 0 = 0, Y 0 = 0.

(5)

Introduction

The two and three dimensional processes Higher Iterations

An extension – any (prime) number of values

Beyond the product (work in progress)

An application, a connection and more

(6)

The two and three dimensional processes

Some basic observations

I (X n ) n and (Y n ) n are strongly dependent, yet W n = (X n , Y n ) behaves like a 2-dimensional random walk. Let W _n ⁰ = (X n , Y _n ⁰ ) with (Y _n ⁰ ) _n is an independent copy of (X _n ) _n .

-50 50 100 150

-50 50 100

-50 50 100 150

-150 -100 -50

50 100 150 200

100 200 300

50 100 150 200

-300 -250 -200 -150 -100 -50

Figure : Two simulations of (W n ) n and (W _n ⁰ ) n

(7)

Some basic observations

5000 10 000 15 000 20 000

50 100 150 200

5000 10 000 15 000 20 000

100 200 300

50 100 150 200

100 200 300

Figure : (X _n ) _n ^deter −→ (Y _n ) _n ^deter −→ (W _n ) _n

(8)

Some basic observations

I Locally, (W n ) n ’s behaviour is different to that of a

2-dimensional random walk (with independent components).

æ

ç ç

-1.0 -0.5 0.5 1.0

-1.0 -0.5 0.5 1.0 z=H0,0L & n=1

æ

ç ç

-2.0 -1.5 -1.0 -0.5

-2.0 -1.5 -1.0 -0.5 z=H-1,-1L & n=1

-10 -5 0 5 10

Figure : Possible positions for W ₁ and W ₁₂

(9)

Some basic observations

I (X n ) n and (Y n ) n are both simple symmetric random walks.

I If P(ξ n = 1) = p 6= 1/2, then

I

η n d

6= ξ n :

P(η n = 1) = 1

2 (1 + (2p − 1) ⁿ );

I

η ₁ , . . . , η _n are not independent:

P(η n = ε _n |η _n = ε ₁ , . . . , η _n−1 = ε _n−1 )

= P(η n = ε n |η n−1 = ε n−1 ) =

p

1 − p

(1+ε

n−1

ε

n

)/2

.

I E[X n Y _n ] =

n

X

k,l =1

E[ξ k η _l ] = 1 + X

(k,l )6=(1,1)

E[ξ k η _l ] = 1.

(10)

The Markov property

I W _n is a time-inhomogeneous Markov process:

( X _n+1 = X _n + ξ _n+1

Y _n+1 = Y _n + (−1)

^n−Xn²

ξ _n+1 .

I But, with K n = n mod 4, (X n , Y n , K n ) is a time-homogeneous Markov process:







X _n+1 = X _n + ξ _n+1

Y n+1 = Y n + (−1)

^Kn−Xn²

ξ n+1

K n+1 = K n + 1 mod 4

(11)

The probability of return to (0, 0) and recurrence

Proposition

P (W 4n = (0, 0)) = 2n − 1 n

2n n

1 2

4n

∼ 1

4πn and

P (W 4n+2 = (0, 0)) = 2n + 1 n + 1

2n n

1 2

4n+2

∼ 1 4πn . It follows that

∞

X

n=0

P (W 4n = (0, 0)) = +∞.

Theorem

W _n = (X _n , Y _n ) is recurrent.

(12)

The transition probabilities

Proposition

If ¹ ₂ (n − k) is even,

P (X n = k, Y n = l ) =

n+l 2 n+k+2l

4 n−l −2 2 n+k−2l

4 1 2

n

If ¹ ₂ (n − k) is odd,

P (X n = k, Y _n = l ) =

n+l 2 n+k+2l

4 n−l −2 2 n+k−2l

4 1 2

n

(13)

The three-dimensional process

I Consider the three-dimensional random walk (X n , Y n , Z n ), also denoted W n , where

Z n =

n

X

k=1

ζ k , n ≥ 1 and Z 0 = 0,

I ζ _k =

k

Y

j =1

η _j =

k

Y

j =1 j

Y

i =1

ξ _i :

ζ 1 = ξ 1 ζ 2 = ξ 1 ξ 2 ζ 3 = ξ 1 ξ 2 ξ 3

ζ ₄ = ξ ₁ ξ ₂ ξ ₃ ξ ₄ ζ ₅ = ξ ₁ ξ ₂ ξ ₃ ξ ₄ ξ ₅ ζ ₆ = ξ ₁ ξ ₂ ξ ₃ ξ ₄ ξ ₅ ξ ₆

(14)

The probability of return to (0, 0, 0) and recurrence

Proposition

1. For any n ≥ 2,

P (W 4n = 0) = 2 ⁻⁴ⁿ

n−2

X

k=0

n − 1 k

n k + 1

n − 1 k + 1

and

P (W 4n+2 = 0) = 2 ^−(4n+2)

n

X

k=1

n + 1 k

n − 1 k − 1

n k

n + 1 k + 1

.

2. P(W 2n = 0) = O(n ^α−2 ), for any α ∈ (1/2, 1).

3. (W _n ) _n is transient; it will visit the origin finitely often.

(15)

A functional central limit theorem

I Donsker: X n (t) = 1

√ n X _[nt] converges weakly to a Brownian motion (t ∈ [0, 1]).

I The same is true for Y n (t) = 1

√ n Y _[nt] and Z n (t) = 1

√ n Z _[nt] .

I The functional dependence between X _n (t), Y _n (t) and Z _n (t) is complete lost at infinity.

Theorem

W _n (t) = (X n (t), Y n (t), Z n (t)) converges weakly to a three-

dimensional Brownian motion.

(16)

Bootstrap random walks Higher Iterations

Introduction

The two and three dimensional processes Higher Iterations

An extension – any (prime) number of values

Beyond the product (work in progress)

An application, a connection and more

(17)

The model setting

I η 0,n = ξ n and η 1,n = η n = Q n k=1 ξ _k .

I η _2,n = Q n

k=1 η _1,k = Q n k=1

Q k j =1 ξ _j

.

I η 3,n = Q n

k=1 η _2,k = Q n

`=1

Q ` k=1

Q k j =1 ξ j

.

κ = 0 κ = 1 κ = 2 κ = 3





 ξ 1

ξ 2

ξ ₃ ξ ₄ ξ 5

ξ ₆













ξ 1

ξ 1 ξ 2

ξ ₁ ξ ₂ ξ ₃ ξ ₁ ξ ₂ ξ ₃ ξ ₄ ξ 1 ξ 2 ξ 3 ξ 4 ξ 5

ξ ₁ ξ ₂ ξ ₃ ξ ₄ ξ ₅ ξ ₆











 ξ 1

ξ 2

ξ ₁ ξ ₃ ξ ₂ ξ ₄ ξ 1 ξ 3 ξ 5

ξ ₂ ξ ₄ ξ ₆











 ξ 1

ξ 1 ξ 2

ξ ₂ ξ ₃ ξ ₃ ξ ₄ ξ 1 ξ 4 ξ 5

ξ ₁ ξ ₂ ξ ₅ ξ ₆







ξ ² _n = (±1) ² = 1. Need to understand which ξ’s are “switched on”.

(18)

The model setting

I η 0,n = ξ n and η 1,n = η n = Q n

`=1 ξ _` .

I η κ+1,n = Q n

`=1 η κ,` .

I η _κ,n =

n

Y

`=1

ξ _n−`+1 ^ν

^κ,`

, where ν _κ,` is a deterministic array and

I

ν 0,1 = 1 and ν 0,n = 0 for n ≥ 2;

I

ν _κ,1 = 1 for κ ≥ 0;

I

ν _κ+1,n+1 = ν _κ+1,n + ν _κ,n+1 mod 2.

I For κ ≥ 1 and n ≥ 1,

ν _κ,n = n + κ − 2 n − 1

mod 2.

(19)

The binomial connection (Sierpinski triangle)

1 10 20 30 40

1

10

20

30

40

1 10 20 30 40

1

10

20

30

40

1 10 20 30 40

1

10

20

30

40

1 10 20 30 40

1

10

20

30

40

Figure : A graphical representation of η κ,n (left) and ν κ,n (right)

(20)

A multi-dimensional extension

I Y 0,0 = X 0 = 0 and Y 0,n = X n = P n

`=1 ξ ` .

I Y κ,0 = 0 and Y κ,n = P n

`=1 η κ,` .

I Y _κ,n is a simple symmetric random walk.

I (X _n , Y _κ,n ) “behaves” like a two-dimensional random walk.

I In fact, W _κ,n = (X _n , Y _1,n , . . . , Y _κ,n ) “behaves” like a

(κ + 1)-dimensional random walk.

(21)

A multi-dimensional extension

1 20 40 60 80 100

1

20

40

60

80

100

1 20 40 60 80 100

1

20

40

60

80

100

1 20 40 60 80 100

1

20

40

60

80

100

1 20 40 60 80 100

1

20

40

60

80

100

Figure : A graphical representation of W _κ,n (left) and a

multi-dimensional random walk (right)

(22)

A second functional central limit theorem

Theorem (Lucas, 1878) A binomial coefficient n

m

is divisible by a prime p if and only if at least one of the base p digits of m is greater than the corresponding digit of n.

I 73 25

= 23214764053299962052 is even: 73 1001001 2

25 0011001 2

.

I For example, for κ = 2 ^ω and 2 ≤ n ≤ 2 ^ω , then ν κ,n = 0.

Theorem

W _κ,n (t) = (X n (t), Y 1,n (t)), . . . , Y κ,n (t)) converges weakly to a

(κ + 1)-dimensional Brownian motion.

(23)

Introduction

The two and three dimensional processes Higher Iterations

An extension – any (prime) number of values

Beyond the product (work in progress)

An application, a connection and more

(24)

An extension – any (prime) number of values

Three-value model

I Suppose (ξ _n ) _n i.i.d. uniform on U = {u, a, b}.

I In general, ab / ∈ U . Therefore ξ ₁ ξ 2 d

6= ξ ₂ .

I How do we recycle in this case?

I Repace × with ⊗:

⊗ u a b

u u a b

a a b u

b b u a

I Observe that u ^⊗3 = a ^⊗3 = b ^⊗3 = u.

I If η n = N n

`=1 ξ ` then (η n ) n

= (ξ d n ) n .

I This can be repeated and generalised.

(25)

General case

I (ξ _n ) _n i.i.d. uniform on U = {u ₀ , u ₁ , . . . , u _p−1 }, where p is prime and P p−1

i =0 u _i = 0.

I Form an Abelian (cyclic) group (U , ⊗).

I Define η κ,n : η 1,n =

n

O

`=1

ξ ` and η κ,n =

n

O

`=1

η κ−1,` =

n

O

`=1

ξ _n−`+1 ^⊗ν

^κ,`

with ν κ,n = n + κ − 2 n − 1

mod p.

I (η _κ,n ) _n has the same distribution as (ξ _n ) _n .

I Let Y _κ,n =

n

X

`=1

η _κ,` , with Y _κ,0 = 0.

(26)

An extension – any (prime) number of values

General case

Proposition

The vector (η 0,n+κ , . . . , η κ,n+κ ) is uniform over U ^κ+1 and is inde- pendent of F _n−1 . In particular, η _0,n+κ , . . . , η _κ,n+κ are independent.

Let σ ² = E[ξ ² n ] = 1

p (u ² ₀ + · · · + u _p−1 ² ) and Y _{K ,n} (t) = 1 σ √

n Y _{K ,bntc} . Theorem

W _κ,n (t) = (X _n (t), Y _1,n (t)), . . . , Y _κ,n (t)) converges weakly to a (κ + 1)-dimensional Brownian motion.

I What happens when p is not prime?

I We add sufficiently many zeroes...

(27)

Introduction

The two and three dimensional processes Higher Iterations

An extension – any (prime) number of values

Beyond the product (work in progress)

An application, a connection and more

(28)

Beyond the product (work in progress)

Representing (φ _n ) _n

I Let η _n = φ _n (ξ ₁ , . . . , ξ _n ). Look for (φ _n ) _n s.t. (η _n ) _n = (ξ ^d _n ) _n .

1 2 3

4 5 6 7

8 9 10 11 12 13 14 15

x ₁ x ₂ x ₃

Figure : The tree representation of the functions φ n .

(29)

Representing (φ _n ) _n

I Let η _n = φ _n (ξ ₁ , . . . , ξ _n ). Look for (φ _n ) _n s.t. (η _n ) _n = (ξ ^d _n ) _n .

1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

Ξ

₁

Ξ

₂

Ξ

3

Φ

1

Figure : The tree representation of the functions φ n .

(30)

Representing (φ _n ) _n

I Let η _n = φ _n (ξ ₁ , . . . , ξ _n ). Look for (φ _n ) _n s.t. (η _n ) _n = (ξ ^d _n ) _n .

1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

Ξ

1

Ξ

₂

Ξ

₃

Φ

1

Φ

₂

Figure : The tree representation of the functions φ n .

(31)

Representing (φ _n ) _n

I Let η _n = φ _n (ξ ₁ , . . . , ξ _n ). Look for (φ _n ) _n s.t. (η _n ) _n = (ξ ^d _n ) _n .

1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

Ξ

1

Ξ

₂

Ξ

₃

Φ

1

Φ

₂

Φ

₃

Figure : The tree representation of the functions φ n .

(32)

Describing (φ _n ) _n

Theorem

Let η n = φ n (ξ 1 , . . . , ξ n ), then (η n ) n d

= (ξ n ) n if and only if φ _n (x ₁ , . . . , x _n ) is of the following form:

φ ₁ (x ₁ ) = (−1) ^α

¹

x ₁

and for n ≥ 2, with K(n), the set of all non-empty subsets of {1, . . . , n − 1},

φ n (x 1 , . . . , x n ) = (−1) ^α

ⁿ



 Y

K ∈K(n)

x _{[K ]} ^β

^n,K



 x n ,

where x _{[K ]} = max _k∈K x _k and α _n , β _n,K ∈ {0, 1}.

(33)

An example

η ₁ = ξ ₁ , η ₂ = ξ ₂ ξ ₁ and η _n = max(ξ _n−2 , ξ _n−1 )ξ _n for n ≥ 3.

1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Figure : The tree representation of the functions max(x n−2 , x n−1 )x n . Theorem

(X _n (t), Y _n (t))) converges weakly to a correlated Brownian motion.

(34)

An example

-500 -400 -300 -200 -100

-150 -100 -50 50 100

-500 -400 -300 -200 -100

-400 -300 -200 -100

-500 -400 -300 -200 -100

-400 -300 -200 -100

-500 -400 -300 -200 -100 -50

50 100 150 200

-500 -400 -300 -200 -100

-400 -300 -200 -100

-500 -400 -300 -200 -100

-400 -300 -200 -100

-500 -400 -300 -200 -100

-200 -100 100

-500 -400 -300 -200 -100

-400 -300 -200 -100

-500 -400 -300 -200 -100

-400 -300 -200 -100

Figure : (W n ) n for φ n (x 1 , . . . , x n ) = max(x n−k , . . . , x n−1 )x n .

(35)

Introduction

The two and three dimensional processes Higher Iterations

An extension – any (prime) number of values

Beyond the product (work in progress)

An application, a connection and more

(36)

An application, a connection and more

A possible application – a greedy random bits generator

1 20 40 60 80 100

1

20

40

60

80

100

1 20 40 60 80 100

1

20

40

60

80

100

100 bits

1 20 40 60 80 100

1

20

40

60

80

100

1 20 40 60 80 100

1

20

40

60

80

100

200 bits

Figure : Simulate or compute?

(37)

A possible application – a greedy random bits generator

1 20 40 60 80 100

1

20

40

60

80

100

1 20 40 60 80 100

1

20

40

60

80

100

300 bits

1 20 40 60 80 100

1

20

40

60

80

100

1 20 40 60 80 100

1

20

40

60

80

100

10000 bits

Figure : Simulate or compute?

(38)

A connection – cellular automata

I Model could be regarded as a long-range cellular automaton.

Figure : Elementary cellular automata

(39)

A connection – cellular automata

I Model could be regarded as a long-range cellular automaton.

Figure : Elementary cellular automata

(40)

A connection – cellular automata

I Up to a “sliding” of the columns, it is also equivalent to CA60.

1 10 20 30 40

1

10

20

30

40

1 10 20 30 40

1

10

20

30

40

1 10 20 30 40

1

10

20

30

40

1 10 20 30 40

1

10

20

30

40

Figure : Cellular automaton 60

(41)

A related question – percolation (work in progress)

1 10 20 30 40

1

10

20

30

40

1 10 20 30 40

1

10

20

30

40

Figure : p = 0.5

(42)

A related question – percolation (work in progress)

1 10 20 30 40

1

10

20

30

40

1 10 20 30 40

1

10

20

30

40

Figure : p = 0.5

(43)

A related question – percolation (work in progress)

1 10 20 30 40

1

10

20

30

40

1 10 20 30 40

1

10

20

30

40

Figure : p = 0.5

(44)

A related question – percolation (work in progress)

1 200 400 600 800

1

200

400

600

800

1 200 400 600 800

1

200

400

600

800

Figure : p = 0.5

(45)

A related question – percolation (work in progress)

1 200 400 600 800 1000

1

200

400

600

800

1000

1 200 400 600 800 1000

1

200

400

600

800

1000

Figure : p = 0.99

(46)

Bootstrap random walks