Different scaling regimes for geometric Poisson functionals

Transcription

1 Different scaling regimes for geometric Poisson functionals Raphaël Lachièze-Rey, Univ. Paris 5 René Descartes Joint work with Giovanni Peccati, University of Luxembourg isson Point Processes: Malliavin Calculus, Wiener-Ito Chaos Expansions and Stochastic Geometry

2 Framework η λ : Poisson measure with intensity µ λ. Chaos decomposition: F λ = F (η λ ) = k q=1 F q,λ, L 2 variable for each λ. Asymptotic regime of renormalized variables F λ = F λ EF λ var(fλ ) F q,λ = F q,λ EF q,λ var(fq,λ ) N : Standard law P(c): Poisson law with parameter c d W : Wasserstein distance.

3 Finite decompositions and U-statistics Under proper integrability assumptions (η: Poisson point process): k-th order stochastic integral with kernel f : I k (f ) = f (x k )d(η µ) k. X k k-th order U-statistic with kernel h: U k (h) = h(x k ) = h(x k )dη k (x k ). x k η X k λ Each k-tuple of points gives a contribution independent of the other points of η λ.

4 Geometric framework η λ : Marked Poisson process. l: Lebesgue measure X λ = [ λ 1/d, λ 1/d ] (M, ν): Marks probability space x = (t, m): marked point. µ λ = 1 Xλ l ν (Lebesgue measure). Kernel scale change Equivalent to µ λ = 1 X1 λl ν.

5 Graph model with unbounded connections H λ R d measurable. x, x η connected if x x H λ. If H λ = Unit ball Unit disk graph ( Boolean model). F λ : Number of connections. Interaction volume: v λ := l(h λ X λ ). v λ 2λ 1/d

6 Different regimes: H λ = α λ H 1 H 1 : a = 1, α λ = 1, λ = 25: y 1/y a 1/x a x CLT at speed λ 1/2 a = 1, α λ = λ 1/d, λ = 50: CLT in log(λ) 1 a = 2, α λ = λ 1/d, λ = 50: no CLT.

7 (R0) λv λ is bounded. Theorem (R1)v λ 0 and λv λ. (R2) v λ c > 0 (R3) v λ. Assume O-regular variation: 0 < c 1 v λ v cλ c 2 < for c > 0. There are C i, C i > 0 such that (R3) : var(f λ ) C 3 λvλ 2 CLT: d W ( F λ, N ) C 3 λ 1/2 (R2) var( F λ, N ) C 2 λ (Sandard behaviour) CLT: d W ( F λ, N ) C 2 λ 1/2 (R1) var(f λ (H)) C 1 λv λ CLT: d W ( F λ, N ) C 1 (λv λ) 1/2 (R0) F λ converges to a Poisson law (or to 0).

8 Hierarchy of chaoses As v λ grows, F 1,λ becomes more predominant. Under (R3) the first chaos dominates. Under (R2) Under (R1),(R0) the second chaos dominates. Remark: CLT var(f λ ). var(f 1,λ ) var(f 2,λ ), 0 < c var(f 1,λ) var(f 2,λ ) C < var(f 2,λ ) var(f 1,λ )

9 Summary: (R3) Large interactions: CLT, first chaos dominates. (R2) Constant size interactions: CLT, chaoses co-dominate. (R1) Small interactions: Slow CLT, 2d chaos dominates. (R0) Rare interactions: Poisson limit, 2d chaos dominates.

10 Subgraph counting G: connected formal graph with cardinality k 1. η λ : Homogeneous Poisson process on X λ. G λ : Graph obtained by connecting points (x, y) η λ with distance x y α λ. v λ := α d λ. F λ (G): Number of occurences of G as a subgraph of G λ.

11 (R1)v λ 0 and λv k 1 λ. (R2) v λ c > 0 (R3) v λ. Results: Penrose, Peccati, LR. (R1) var(f λ ) c 1 λv k 1 λ CLT: d W ( F λ, N ) C 1 (λv k 1 λ ) 1/2 (R2) var(f λ ) c 2 λ CLT: d W ( F λ, N ) C 2 λ 1/2 (R3) var(f λ ) c 3 λv 2k 2 λ CLT: d W ( F λ, N ) C 3 λ 1/2

12 Stationary rescaled marked kernels k 2. F λ = U k (h λ ) = Assumptions on h λ : y 1,...,y k η λ h λ (y 1,..., y k ), y i = (t i, m i ) R d M. h λ ( ) h(α λ ) in some sense. α λ : Scaling regime. ( v λ := αλ d : Interaction measure. ) h(y 1,..., y k ) invariant under spatial translations. h is rapidly decreasing away from the diagonal: there exists κ(y) > 0 bounded probability density such that for p = 2, 4, M (Rd M)k 1 h(0, m 0, y 1,..., y k 1 ) p (κ(y 1 )... κ(y k 1 )) p 1 µ ν(dy 1,..., dy k 1 ) <.

13 Tool: Bounds on the contractions Theorem For F = k q=1 I q(f q ) L 2 with var(f ) = 1, d W (F, N ) C(k)(max f q l r f q 2 + max f q 4 ) q where 1 l r q q and l q f q l r f q (x r l, y q r, y q r ) = f q (x l, x r l, y q r )f q (x l, x r l, y q r )dx l. Theorem If h(x k ) and g(x q ) are stationary and rapidly decreasing, for r, l as above, h l r g 2 L 2 (X (R d ) k+q r l 1 ) l(x )A(h)A(g)

14 Chaos behaviour F λ = U k (h λ ) = Theorem q-th chaos behaviour of U k (h λ ): k F q,λ. q=1 var(f q,λ ) Cλv 2k q 1 λ { d W ( F q,λ, N ) C v 1 q λ v 1 q λ + 1 if v λ > 1 λ {q 1} v λ otherwise First term: kernel 4-th moments. second term: kernel contractions. First chaoses win if v λ. Last chaoses win if v λ 0. High order chaoses convergence is slower.

15 Applications v λ = λ: All points interact Geometric U-statistics. v λ = 1: Standard behaviour/thermodynamic regime. v λ small: rarefaction of interactions: Slow CLT/no CLT.

16 Examples of functionals with a standard behaviour Boolean model: M: Compact sets (with Fell Borel σ-algebra). {M k ; k 1} IID Random compact sets. {x k ; k 1 } Poisson point process with intensity λ. η λ = {(x k, M k )} marked Poisson measure. R λ = (M k x k ) k:x k X λ F λ : U-statistic with stationary kernel h((x, M); (x, M )) = ϕ(x x )1 {(M x) (M x ) }, F λ = ϕ(x i x j )1 {The grains with centers xi and x j touch}. x i x j η λ

17 Magnitude assumption on ϕ ϕ(x y) x y β, x, y R d. Theorem Assume that in the boolean model the typical grain has diameter R such that for some ε > 0, β > d/2, and for r 1 then for some C, C > 0 P(R r) Cr (2(β+d)+1+ε) var(f λ ) Cλ d W ( F λ, N ) C λ 1/2. The optimal condition bears actually upon the decay of χ(x) = P(M 1 (M 2 x) ), x R d.

18 Number of intersections in a process of line segments. M k ; k 1 Line segments with random IID lengths. ϕ(x, y) = 1. Then F λ is the number of intersections of segments with centers in X λ. Figure: Zbynek Pawlas Theorem (Pawlas 2012 ) There is standard behaviour if for some ε > 0 P(length(M 1 ) r) Cr 5 ε, r 1.

19 sub-hypergraph counting and telecommunications network Decreusefond et al. η λ = {x k ; k 1} Poisson point process with intensity λl on the torus. k-th order hypergraph H k,λ : Data of k-tuples (x 1,..., x k ) η λ such that x i x j ε Exact mean formulas for H k,λ and related topological quantities. Our approach: {m k ; k 1} random radii around the points. Edge effects: η λ = {y k = (x k, m k ); k 1} on X λ = [ λ 1/d, λ 1/d ]. H k,λ : k-tuples (y 1,..., y k ) such that for all i j, x i x j m i. Results(R=typical radius) : Theorem If ER 4d+ε <, var(f k,λ ) Cλ d W ( H k,λ, N ) C λ 1/2

20 Geometric U-statistics: α λ = λ 1/d Points interact regardless of the distance: F λ = y k η λ h(λ 1/d y 1,..., λ 1/d y k ). General form of a Geometric U-statistic: (X, µ) loc. compact measured space µ λ = λµ h(y 1,..., y k ): kernel on X k. F λ = y k η λ h(x 1,..., x k ). Examples[Reitzner and Schulte] Number of k-tuples of points in convex position Approximation of Sylvester s constant in a convex body. Intersections of flats in a compact window.

21 Results k F λ = κ k,q F q,λ q=1 and F q,λ = λ k i I q (h q ) Kernel projections: h q (y q ) = h(y q, y k q )dµ(y q ) X q Asymptotic behaviour: Theorem var(f q,λ ) Cλ 2k q F q,λ G q (h q ) where G q (h q ) is a Gaussian chaos of order q.

22 Summary for geometric U-statistics F λ behaves like F q0,λ. q 0 = min{q : h q 2 0}. Reitzner and Schulte. q 0 = 1 CLT at speed λ 1/2 LR and Peccati: q 0 2 no CLT, convergence to G q0 (h q0 ). Peccati and Thaele: q 0 = 2: Speed of convergence to G 2, a Γ random variable.

23 Poisson regime: λα λ c Peccati (2011) : sufficient conditions for the convergence to a Poisson law in terms of the contractions.

24 Mixed chaos behaviour Multi-dimensional CLTs: Peccati, Zengh, Minh, Schulte, Thaele, Last, Penrose, Reitzner, LR... Peccati, Bourguain 2012: Portmanteau inequalities Mixed limit theorems. Example. Disk graph with influence volume v λ such that λv 3 1 λ 0 and. Consider λv 2 1 λ Then where N and P are independent. F 2,λ = #segments F 3,λ = #triangles. ( F 2,λ, F 3,λ ) (N, P)