Patterns in random walks and Brownian motion CSP conference in honor - - PowerPoint PPT Presentation

Patterns in random walks and Brownian motion

CSP conference in honor of J. Pitman 20 − 21 June 2014, UC San Diego Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman 20 June, 2014

Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion

Scenario #1

Question Given some distribution of a process X with continuous paths, is there a random time T such that (BT+u − BT; 0 ≤ u ≤ 1) has the same distribution as (Xu, 0 ≤ u ≤ 1) ?. Examples : Brownian/pseudo bridge, Brownian meander, normalized excursion, Bessel(3), Vervaat bridges. . . etc. The question here has some affinity to the well-known Skorokhod embedding problem. The question is related to splitting theorems of post-T Markov processes.

Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion

Scenario #2

Question Given a Borel measurable subset S ⊂ C[0, 1], can we find a random time T such that (BT+u − BT; 0 ≤ u ≤ 1) ∈ S with probability one ? Examples : E := {w ∈ C[0, 1]; w(t) > w(1) = 0 for 0 < t < 1}; M := {w ∈ C[0, 1]; w(t) > 0 for 0 < t ≤ 1}; BRλ := {w ∈ C[0, 1]; w(1) = λ}; FPλ := {w ∈ C[0, 1]; w(t) > w(1) = λ for 0 ≤ t < 1}; VBλ := {w ∈ FPλ; ζ := inf{t > 0; w(t) < 0} > 0} . . . etc.

Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion

Scenario #3

Question Given for each n a collection An of patterns of length n, what is the order of the expected waiting time ET(An) until one of the elements of An is observed in a random walk ? Examples : E2n := {w ∈ SW(2n); w(i) > 0 for 1 ≤ i ≤ 2n−1 and w(2n) = 0}; M2n+1 := {w ∈ SW(2n + 1); w(i) > 0 for 1 ≤ i ≤ 2n + 1}; BRλ,n := {w ∈ SW(n); w(n) = λn} where λn ∼ λ √ n; FPλ,n := {w ∈ SW(n); w(i) > w(n) = λn for 0 ≤ i ≤ n−1} . . . etc.

Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion

Response to Scenario #3

Theorem

1

There exists CE > 0 such that ET(E2n) ∼ CEn

3 2 ; 2

There exists CM > 0 such that ET(M2n+1) ∼ CMn;

3

There exists Cλ

BR > 0 such that

ET(BRλ,n) ∼ Cλ

BRn;

4

There exists cλ

FP and Cλ FP > 0 such that

cλ

FPn ≤ ET(FPλ,n) ≤ Cλ FPn

5 4 . Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion

Response to Scenario #2

Theorem

1

a.s. ∄ random time T such that (BT+u − BT; 0 ≤ u ≤ 1) ∈ E := {w ∈ C[0, 1]; w(t) > w(1) = 0 for 0 < t < 1};

2

a.s. ∄ random time T such that (BT+u − BT; 0 ≤ u ≤ 1) ∈ RBR := {w ∈ C[0, 1]; w(t) ≥ w(1) = 0 for 0 ≤ t ≤ 1};

3

For each λ < 0, a.s. ∄ T s.t. (BT+u − BT; 0 ≤ u ≤ 1) ∈ VBλ := {w ∈ FPλ; ζ := inf{t > 0; w(t) < 0} > 0}, where FPλ := {w ∈ C[0, 1]; w(t) > w(1) = λ for 0 ≤ t < 1}. Consequence : no normalized excursion, reflected bridges, Vervaat bridges in a Brownian path ! Idea : Williams’ path decompositions, or fragmentation argument.

Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion

Response to Scenario #1

Theorem For each of the following three processes X := (Xu, ≤ u ≤ 1) there is some random time T such that (BT+u − BT; 0 ≤ u ≤ 1) has the same distribution as X :

1

the meander X = (mu; 0 ≤ u ≤ 1) ;

2

the co-meander X = ( mu; 0 ≤ u ≤ 1) ;

3

the Bessel(3) process X = (Ru; 0 ≤ u ≤ 1). Idea : Brownian meander by Itˆ

• ’s excursion theory and the
• ther two by acceptance-rejection method.

Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion

Our favorite open problem

Open problem Can we find a random time T such that (BT+u − BT; 0 ≤ u ≤ 1) has the same distribution as Brownian bridge (b0

u; 0 ≤ u ≤ 1) ?

The bridge pattern BR0 is achieved by the bridge-like process (BT+u − BT; 0 ≤ u ≤ 1), where T := inf{t > 0; Bt − Bt+1 = 0}. The bridge-like process can be inferred from the work of Slepian and Shepp = ⇒ P(T > t) is computed. From simulation, the above bridge-like process is not Brownian bridge. The related Slepian zero set has rich properties, work in progress with Jim Pitman.

Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion

Remerciement

Thank you for your attention, AND Happy birthday, Jim !

Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion