Convex hulls of random walks Andrew Wade Department of - PowerPoint PPT Presentation

Convex hulls of random walks ● Andrew Wade Department of Mathematical Sciences Durham University March 2014 Joint work with Chang Xu, University of Strathclyde

Introduction On each of n unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, what is the minimum length of fencing needed to enclose the garden? ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀

Introduction Let Z 1 , Z 2 , . . . be independent, identically distributed random vectors in R 2 . The Z i will be the increments of the planar random walk S n , n ≥ 0, defined by S 0 = 0 (the origin in R 2 ) and n � S n = Z i . i = 1 We are interested in asymptotic properties of the convex hull hull ( S 0 , . . . , S n ) . In particular, the n → ∞ limit behaviour of the random variables L n = the perimeter length of hull ( S 0 , . . . , S n ) .

Introduction Standing assumption: E ( � Z 1 � 2 ) < ∞ (two moments). There is going to be a clear distinction between the zero drift case ( E Z 1 = 0) and the non-zero drift case ( � E Z 1 � > 0). For example, under mild conditions: • the zero-drift walk is recurrent and the convex hull tends to the whole of R 2 ; • the walk with drift is transient with a limiting direction and the convex hull sits inside some arbitrarily narrow wedge. We will look at both cases in this talk. First, we give a brief summary of some history.

Introduction 1 Background 2 Zero drift and Brownian scaling limit 3 Non-zero drift and central limit theorem 4 Concluding remarks 5

Some history Spitzer & Widom (1961) and Baxter (1961) showed that n 1 � E L n = 2 k E � S k � . k = 1 So, under mild conditions: • the zero-drift case has E L n ≍ √ n ; • the case with drift has E L n ≍ n . Snyder & Steele (1993) showed that n V ar ( L n ) ≤ π 2 1 � E � Z 1 � 2 − � E Z 1 � 2 � . (1) 2 Snyder & Steele deduced from (1) the strong law n →∞ n − 1 L n = 2 � E Z 1 � , a.s. lim

Some questions The work of Snyder & Steele raised some natural questions. • Is n the correct order for V ar ( L n ) ? • Is there a distributional limit theorem for L n ? • If so, is the limit distribution normal? The answers to these questions turn out be be essentially • yes, yes, no in the zero drift case, and • yes, yes, yes in the non-zero drift case, excluding some degenerate cases. First, we have a quick look at some simulation evidence.

Some simulations As a concrete example, let Z 1 be distributed as ( µ, 0 ) + e Θ , where • e θ = ( cos θ, sin θ ) is the unit vector in direction θ ; • Θ is a uniform random variable on [ 0 , 2 π ) . So E Z 1 = ( µ, 0 ) , � E Z 1 � = µ . The case µ = 0 is the Pearson–Rayleigh random walk. Picture for µ = 0 . 2:

Some simulations: non-zero drift • Left: Histogram of L n samples for n = 10 4 ; 10 6 simulations. • Right: Plot of point estimates for y = V ar ( L n ) against x = n ; also shown is the line y = 2 x . 70000 60000 50000 ● ● ● 4e+05 ● ● 40000 ● ● ● ● ● ● 30000 ● ● 2e+05 ● ● 20000 ● ● ● ● ● 10000 ● 0e+00 ● ● ● ● ● ● 0 0 50000 150000 250000 200 300 400 500 600 700 Simulations suggest: n − 1 V ar ( L n ) → const. (2 in this case!) ; L n satisfies a CLT .

Some simulations: zero drift • Left: Histogram of L n samples for n = 10 4 ; 10 6 simulations. • Right: Plot of point estimates for y = V ar ( L n ) against x = n ; also shown is the line y = 0 . 536 x . 4e+05 2e+05 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0e+00 ● ● ● ● ● ● 0 50000 150000 250000 100 150 200 250 300 350 Simulations suggest: n − 1 V ar ( L n ) → const. ≈ 0 . 536 ; L n non-Gaussian limit .

First tool: Cauchy formula Let e θ = ( cos θ, sin θ ) , unit vector in direction θ . Set M n ( θ ) = max 0 ≤ k ≤ n ( S k · e θ ) , m n ( θ ) = min 0 ≤ k ≤ n ( S k · e θ ) . Cauchy’s perimeter formula from convex geometry: � π L n = ( M n ( θ ) − m n ( θ )) d θ. 0 M n ( θ ) θ

First tool: Cauchy formula � π L n = ( M n ( θ ) − m n ( θ )) d θ. 0 A first consequence: classical fluctuation theory for random walk on R gives n � k − 1 E [( S k · e θ ) + ] , E M n ( θ ) = k = 1 a formula attributed variously to Kac, Hunt, Dyson, and Chung, and which can be proved combinatorially, or analytically as a consequence of the Spitzer–Baxter fluctuation theory identities. Then � π n n � k − 1 E � k − 1 E � S k � , E L n = | S k · e θ | d θ = 2 0 k = 1 k = 1 which is the Spitzer–Widom formula.

Zero drift case Suppose E Z 1 = 0. The random walk has Brownian motion as its scaling limit. So one would expect that the convex hull of the random walk is described in the limit by the convex hull of Brownian motion. The latter was studied by L´ evy; more recently by El Bachir (1983) and others. We need to know a little about convex hulls of continuous ● paths, and need to set things up on the right space(s).

Paths and hulls Consider continuous f : [ 0 , T ] → R d with f ( 0 ) = 0; say f ∈ C 0 d . ( T is not very important—enough to take T ≡ 1.) With the supremum norm ρ ∞ ( f , g ) = sup x � f ( x ) − g ( x ) � we get a metric space ( C 0 d , ρ ∞ ) . The path segment ( ≡ interval image) f [ 0 , t ] = { f ( s ) : s ∈ [ 0 , t ] } is compact. = ⇒ hull ( f [ 0 , t ]) is compact (by a theorem of Carath´ eodory). That is, hull ( f [ 0 , t ]) is an element of the metric space ( K 0 d , ρ H ) of compact convex subsets of R d containing 0, with the Hausdorff metric.

Paths and hulls Metric space ( K 0 d , ρ H ) of compact convex subsets of R d containing 0, with the Hausdorff metric. d and r > 0, let A r := { x ∈ R d : ρ ( x , A ) ≤ r } . Given A ∈ K 0 For A , B ∈ K 0 d , A ⊆ B r and B ⊆ A r . ρ H ( A , B ) ≤ r ⇔ Lemma 1 For each t, the map f �→ hull ( f [ 0 , t ]) is a continuous function from ( C 0 d , ρ ∞ ) to ( K 0 d , ρ H ) .

Scaling limit Given random walk S n = � n i = 1 Z i , define X n ( t ) := n − 1 / 2 � � �� S ⌊ nt ⌋ + ( nt − ⌊ nt ⌋ ) S ⌊ nt ⌋ + 1 − S ⌊ nt ⌋ . So for each n , X n ∈ C 0 d ; X n ( 0 ) = 0 and X n ( 1 ) = n − 1 / 2 S n . Let b t , t ≥ 0 denote standard Brownian motion on R d . Donsker’s Theorem 1 ) = σ 2 I, σ 2 > 0 . Suppose E ( � Z 1 � 2 ) < ∞ , E Z 1 = 0 , and E ( Z 1 Z ⊤ Then X n /σ ⇒ b in the sense of weak convergence on ( C 0 d , ρ ∞ ) . Note hull ( X n [ 0 , 1 ]) = n − 1 / 2 hull ( S 0 , . . . , S n ) . Then with Lemma 1 and the continuous mapping theorem, we get: Theorem 2 Under the same conditions, n − 1 / 2 hull ( S 0 , . . . , S n ) ⇒ hull ( b [ 0 , 1 ]) in the sense of weak convergence on ( K 0 d , ρ H ) .

Functionals Now take d = 2. One neat way to define perimeter length of a set A ∈ K 0 2 is via intrinsic volumes: � | A r | − | A | � L ( A ) := lim , r r ↓ 0 where | · | is Lebesgue measure on R 2 ; the limit exists by the Steiner formula of integral geometry. In particular, � H 1 ( ∂ A ) if int ( A ) � = ∅ L ( A ) = 2 H 1 ( ∂ A ) if int ( A ) = ∅ where H 1 is one-dimensional Hausdorff measure.

Functionals Lemma 3 The map A �→ L ( A ) is a continuous function from ( K 0 2 , ρ H ) to ( R + , ρ ) . Note L ( X n [ 0 , 1 ]) = L ( n − 1 / 2 hull ( S 0 , . . . , S n )) = n − 1 / 2 L n . Corollary 4 1 ) = σ 2 I, σ 2 > 0 . Suppose E ( � Z 1 � 2 ) < ∞ , E Z 1 = 0 , and E ( Z 1 Z ⊤ d Then n − 1 / 2 L n − → ℓ 1 , where ℓ 1 = L ( hull ( b [ 0 , 1 ])) is the perimeter length of the convex hull of planar Brownian motion run for unit time. Assuming E ( � Z 1 � 2 + ε ) < ∞ , a uniform integrability argument gives n →∞ n − 1 V ar ( L n ) = V ar ( ℓ 1 ) . lim Work in progress. We can show V ar ( ℓ 1 ) > 0. We’d like an exact formula. Goldman (1996) manages to do a similar calculation for the planar Brownian bridge, but it is tricky.

Non-zero drift case Now suppose � E Z 1 � > 0. Our results are: Theorem 5 n V ar ( L n ) = 4 E [(( Z 1 − E Z 1 ) · E Z 1 ) 2 ] 1 =: s 2 ∈ [ 0 , ∞ ) . lim � E Z 1 � 2 n →∞ Theorem 6 Suppose that s 2 > 0 . Then for any x ∈ R , � L n − E L n � � L n − E L n � √ V arL n √ lim ≤ x = lim ≤ x = Φ( x ) , n →∞ P n →∞ P ns 2 the standard normal distribution function.