Bootstrap random walks

Transcription

1 Bootstrap random walks Kais Hamza Monash University Joint work with Andrea Collevecchio & Meng Shi

2 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 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 Introduction The model setting Let (ξ n ) n be a sequence of independent identically distributed random variables such that P(ξ i = ) = P(ξ i = +) = 2. Let η k = k j= ξ j. Then (η n ) n d = (ξn ) n. The σ-algebras generated by these two sequences are the same, i.e. σ(ξ,, ξ n ) = σ(η,, η n ). Let X n = n k= ξ k and Y n = n k= η k, with X =, Y =.

5 The two and three dimensional processes 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 (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 = (X n, Y n) with (Y n) n is an independent copy of (X n ) n Figure : Two simulations of (W n ) n and (W n) n

7 The two and three dimensional processes Some basic observations Figure : (X n ) n deter (Y n ) n deter (W n ) n

8 The two and three dimensional processes Some basic observations Locally, (W n ) n s behaviour is different to that of a 2-dimensional random walk (with independent components). z, & n. z, & n Figure : Possible positions for W and W 2

9 The two and three dimensional processes Some basic observations (X n ) n and (Y n ) n are both simple symmetric random walks. If P(ξ n = ) = p /2, then η n d ξn : P(η n = ) = 2 ( + (2p )n ); η,..., η n are not independent: E[X n Y n ] = P(η n = ε n η n = ε,..., η n = ε n ) ( ) (+εn ε p n)/2 = P(η n = ε n η n = ε n ) =. p n E[ξ k η l ] = + k,l= (k,l) (,) E[ξ k η l ] =.

10 The two and three dimensional processes The Markov property W n is a time-inhomogeneous Markov process: { Xn+ = X n + ξ n+ Y n+ = Y n + ( ) n Xn 2 ξ n+. But, with K n = n mod 4, (X n, Y n, K n ) is a time-homogeneous Markov process: X n+ = X n + ξ n+ Y n+ = Y n + ( ) Kn Xn 2 ξ n+ K n+ = K n + mod 4

11 The two and three dimensional processes The probability of return to (, ) and recurrence Proposition ( )( ) ( ) 2n 2n 4n P (W 4n = (, )) = n n 2 4πn and ( )( ) ( ) 2n + 2n 4n+2 P (W 4n+2 = (, )) = n + n 2 4πn. It follows that P (W 4n = (, )) = +. n= Theorem W n = (X n, Y n ) is recurrent.

12 The two and three dimensional processes The transition probabilities Proposition If 2 (n k) is even, P (X n = k, Y n = l) = If 2 (n k) is odd, P (X n = k, Y n = l) = ( n+l 2 n+k+2l 4 ( n+l 2 n+k+2l 4 )( n l 2 2 n+k 2l 4 )( n l 2 2 n+k 2l 4 ) ( ) n 2 ) ( ) n 2

13 The two and three dimensional processes The three-dimensional process Consider the three-dimensional random walk (X n, Y n, Z n ), also denoted W n, where ζ k = k η j = j= Z n = k j= i= n ζ k, n and Z =, k= j ξ i : ζ = ξ ζ 2 = ξ ξ 2 ζ 3 = ξ ξ 2 ξ 3 ζ 4 = ξ ξ 2 ξ 3 ξ 4 ζ 5 = ξ ξ 2 ξ 3 ξ 4 ξ 5 ζ 6 = ξ ξ 2 ξ 3 ξ 4 ξ 5 ξ 6

14 The two and three dimensional processes The probability of return to (,, ) and recurrence Proposition. For any n 2, n 2 ( )( )( )( ) n n n n P (W 4n = ) = 2 4n k k + k + k + k= and n ( )( )( )( ) n + n n n + P (W 4n+2 = ) = 2 (4n+2). k k k k + k= 2. P(W 2n = ) = O(n α 2 ), for any α (/2, ). 3. (W n ) n is transient; it will visit the origin finitely often.

15 The two and three dimensional processes A functional central limit theorem Donsker: X n (t) = n X [nt] converges weakly to a Brownian motion (t [, ]). The same is true for Y n (t) = n Y [nt] and Z n (t) = n Z [nt]. 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 threedimensional Brownian motion.

16 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 Higher Iterations The model setting η,n = ξ n and η,n = η n = n k= ξ k. η 2,n = n k= η,k = n k= ( k j= ξ j ). η 3,n = n k= η 2,k = n l= ( l k= ( k j= ξ j )). κ = κ = κ = 2 κ = 3 ξ ξ ξ ξ ξ 2 ξ ξ 2 ξ 2 ξ ξ 2 ξ 3 ξ ξ 2 ξ 3 ξ ξ 3 ξ 2 ξ 3 ξ 4 ξ ξ 2 ξ 3 ξ 4 ξ 2 ξ 4 ξ 3 ξ 4 ξ 5 ξ ξ 2 ξ 3 ξ 4 ξ 5 ξ ξ 3 ξ 5 ξ ξ 4 ξ 5 ξ 6 ξ ξ 2 ξ 3 ξ 4 ξ 5 ξ 6 ξ 2 ξ 4 ξ 6 ξ ξ 2 ξ 5 ξ 6 ξ 2 n = (±) 2 =. Need to understand which ξ s are switched on.

18 Higher Iterations The model setting η,n = ξ n and η,n = η n = n l= ξ l. η κ+,n = n l= η κ,l. n η κ,n = ξ ν κ,l n l+, where ν κ,l is a deterministic array and l= ν, = and ν,n = for n 2; ν κ, = for κ ; ν κ+,n+ = ν κ+,n + ν κ,n+ mod 2. For κ and n, ( ) n + κ 2 ν κ,n = n mod 2.

19 Higher Iterations The binomial connection (Sierpinski triangle) Figure : A graphical representation of η κ,n (left) and ν κ,n (right)

20 Higher Iterations A multi-dimensional extension Y, = X = and Y,n = X n = n l= ξ l. Y κ, = and Y κ,n = n l= η κ,l. Y κ,n is a simple symmetric random walk. (X n, Y κ,n ) behaves like a two-dimensional random walk. In fact, W κ,n = (X n, Y,n,..., Y κ,n ) behaves like a (κ + )-dimensional random walk.

21 Higher Iterations A multi-dimensional extension Figure : A graphical representation of W κ,n (left) and a multi-dimensional random walk (right)

22 Higher Iterations A second functional central limit theorem Theorem (Lucas, 878) ( ) n A binomial coefficient is divisible by a prime p if and only if at m least one of the base p digits of m is greater than the corresponding digit of n. ( ) = is even: For example, for κ = 2 ω and 2 n 2 ω, then ν κ,n =. Theorem W κ,n (t) = (X n (t), Y,n (t)),..., Y κ,n (t)) converges weakly to a (κ + )-dimensional Brownian motion.

23 An extension any (prime) number of values 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 Suppose (ξ n ) n i.i.d. uniform on U = {u, a, b}. In general, ab / U. Therefore ξ ξ 2 d ξ2. How do we recycle in this case? Repace with : u a b u u a b a a b u b b u a Observe that u 3 = a 3 = b 3 = u. If η n = n l= ξ d l then (η n ) n = (ξn ) n. This can be repeated and generalised.

25 An extension any (prime) number of values General case (ξ n ) n i.i.d. uniform on U = {u, u,..., u p }, where p is prime and p i= u i =. Form an Abelian (cyclic) group (U, ). Define η κ,n : η,n = with ν κ,n = ( l= n + κ 2 n n ξ l and η κ,n = ) mod p. n η κ,l = l= (η κ,n ) n has the same distribution as (ξ n ) n. n Let Y κ,n = η κ,l, with Y κ, =. l= n l= ξ ν κ,l n l+

26 An extension any (prime) number of values General case Proposition The vector (η,n+κ,..., η κ,n+κ ) is uniform over U κ+ and is independent of F n. In particular, η,n+κ,..., η κ,n+κ are independent. Let σ 2 = E[ξ 2 n] = p (u2 + + u 2 p ) and Y K,n (t) = σ n Y K, nt. Theorem W κ,n (t) = (X n (t), Y,n (t)),..., Y κ,n (t)) converges weakly to a (κ + )-dimensional Brownian motion. What happens when p is not prime? We add sufficiently many zeroes...

27 Beyond the product (work in progress) 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 Let η n = φ n (ξ,..., ξ n ). Look for (φ n ) n s.t. (η n ) n d = (ξn ) n. x 2 3 x x Figure : The tree representation of the functions φ n.

29 Beyond the product (work in progress) Representing (φ n ) n Let η n = φ n (ξ,..., ξ n ). Look for (φ n ) n s.t. (η n ) n d = (ξn ) n. Ξ Φ 2 3 Ξ Ξ Figure : The tree representation of the functions φ n.

30 Beyond the product (work in progress) Representing (φ n ) n Let η n = φ n (ξ,..., ξ n ). Look for (φ n ) n s.t. (η n ) n d = (ξn ) n. Ξ Φ 2 3 Ξ 2 Φ Ξ Figure : The tree representation of the functions φ n.

31 Beyond the product (work in progress) Representing (φ n ) n Let η n = φ n (ξ,..., ξ n ). Look for (φ n ) n s.t. (η n ) n d = (ξn ) n. Ξ Φ 2 3 Ξ 2 Φ Ξ 3 Φ Figure : The tree representation of the functions φ n.

32 Beyond the product (work in progress) Describing (φ n ) n Theorem Let η n = φ n (ξ,..., ξ n ), then (η n ) n d = (ξn ) n if and only if φ n (x,..., x n ) is of the following form: φ (x ) = ( ) α x and for n 2, with K(n), the set of all non-empty subsets of {,..., n }, φ n (x,..., x n ) = ( ) αn x n, K K(n) where x [K] = max k K x k and α n, β n,k {, }. x β n,k [K]

33 Beyond the product (work in progress) An example η = ξ, η 2 = ξ 2 ξ and η n = max(ξ n 2, ξ n )ξ n for n Figure : The tree representation of the functions max(x n 2, x n )x n. Theorem (X n (t), Y n (t))) converges weakly to a correlated Brownian motion.

34 Bootstrap random walks Beyond the product (work in progress) An example Figure : (Wn )n for φn (x,..., xn ) = max(xn k,..., xn )xn.

35 An application, a connection and more 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 Figure : Simulate or compute?

37 An application, a connection and more A possible application a greedy random bits generator Figure : Simulate or compute?

38 An application, a connection and more A connection cellular automata Model could be regarded as a long-range cellular automaton. Figure : Elementary cellular automata

39 An application, a connection and more A connection cellular automata Model could be regarded as a long-range cellular automaton. Figure : Elementary cellular automata

40 An application, a connection and more A connection cellular automata Up to a sliding of the columns, it is also equivalent to CA Figure : Cellular automaton 6

41 An application, a connection and more A related question percolation (work in progress) Figure : p =.5

42 An application, a connection and more A related question percolation (work in progress) Figure : p =.5

43 An application, a connection and more A related question percolation (work in progress) Figure : p =.5

44 An application, a connection and more A related question percolation (work in progress) Figure : p =.5

45 An application, a connection and more A related question percolation (work in progress) Figure : p =.99

46 An application, a connection and more References Collevecchio A., Hamza K. & Shi M. (25) Bootstrap Random Walks, arxiv.org, pp.