Scaling properties of Poisson germ-grain models

Ingemar Kaj

Department of MathematicsUppsala Universityikaj@math.uu.se

Reisensburg Workshop, Feb 2007

Purpose

I In spatial stochastic structure,investigate (long-range)dependence over distance

I independently scattered versusfractal dependence

I Use Poisson germ-grain models,to

- generalize known 1-dimresults (M/G/∞)

- unify heavy-tailed andmicroballs models

- understand self-similarityproperties, etc

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Outline

I Poisson germ-grain

I grain size distribution with heavy tails/singular intensities

I random fields

I scaling analysis; zoom-in or zoom-outI three different limit regimes

- Gaussian- intermediate, Poisson- (stable)

I self-similarityI examples

- fractional Brownian field- “fractional Poisson motion”

Presentation based on joint works

Kaj, Leskela, Norros, Schmidt: Scaling limits for random fields withlong-range dependence. Ann. Probab. 35:2, 2007.

Bierme, Estrade, Kaj: About scaling behavior of random ballsmodels. 6th Int. Conf. on Stereology, Spatial Statistics andStochastic Geometry, Prague, June 2006.

Poisson point process

[grains = balls, in this presentation]

I Construct family of grains inRd, {Xj +B(0, Rj)}, builtfrom Poisson point process inRd ×R+.

I Model: Poisson measureN(dx, dr) on Rd ×R+ withintensity measuren(dx, dr) = dxF (dr)

I if F (dr) distribution: eachpoint (x, r) is random ballwith center x, radius r.

I if not: Microballs model

Background

The analog for d = 1 is the M/G/∞model.

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I With heavy-tailed service times such models underaggregation and proper scaling fluctuate as FractionalBrownian Motion (H > 1/2) in the asymptotic limit.

I More generally, three limit regimes: Gaussian, Intermediate,Stable (IK/Gaigalas, Bernoulli 03; IK, Fractals in Eng 05;IK/Taqqu 2007, etc)

I Bierme and Estrade, JAP 2006, consider micro ball model andderives limit which shares properties of FBM with Hurst indexH < 1/2

Model assumptions

Germ-grain intensity n(dx, dr) = dxF (dr), where F (dr) σ-finitemeasure, with∫

R+rd F (dr) < +∞ finite expected “volume”

and, for β 6= d, F (dr) = f(r)dr where

f(r) ∼ Cβ r−β−1 as r → 0d−β =

{r → 0, β < dr →∞, β > d

Yields random configuration of(super-positioned) mass in Rd.Large grains keep the massconfigurations rigid, small grainscause non-trivial noise effects.

Scaling idea

Study fluctuations around the expected mass configuration{zooming-out (many small heavy-tailed grains limit), d < β (< 2d)

zooming-in (not so many large micro-grains limit), β < d

Scaling parameters

Takenλ,ρ(dx, dr) = λdxFρ(dr)

where Fρ is image measure under r → ρr. As{λ→∞, ρ→ 0 zoom-out (β > d)λ→ 0, ρ→∞ zoom-in (β < d)

the relative speed of Poisson intensity λ and scaled radius ρ, determines

the limit regime:

I Gaussian; λρβ →∞I Intermediate; λρβ → constI Stable; λρβ → 0

Interpretation: E(# of balls of radius > 1 which covers a point) ∼ λρβ

Scaling parameters

Takenλ,ρ(dx, dr) = λdxFρ(dr)

where Fρ is image measure under r → ρr. As{λ→∞, ρ→ 0 zoom-out (β > d)λ→ 0, ρ→∞ zoom-in (β < d)

the relative speed of Poisson intensity λ and scaled radius ρ, determines

the limit regime:

I Gaussian; λρβ →∞I Intermediate; λρβ → constI Stable; λρβ → 0

Interpretation: E(# of balls of radius > 1 which covers a point) ∼ λρβ

Random fields

Pointwise field:

W (y) =∫Rd

∫ ∞

0

1{|y−x|<r}N(dx, dr) = # grains containing y

with

EW (y) =∫Rd

∫ ∞

0

1{|y−x|<r} dxF (dr) = |B(y, 1)|∫Rd

rd F (dr) <∞

More general indexing tool: for measures µ, consider

µ 7→W (µ) =∫Rd

∫ ∞

0

µ(B(x, r))N(dx, dr).

Stochastic integral is well defined as limit in prob. of elementary integrals.Which measures µ?

Fluctuation field

The centered fluctuations are described by the compensated stochasticintegral

Yλ,ρ(µ) =∫Rd

∫ ∞

0

µ(B(x, r)) (Nλ,ρ(dx, dr)− λdxFρ(dr))

Limit problem:

Find normalization n(λ, ρ) such that

Yλ,ρ(µ)/n(λ, ρ) d⇒ Y (µ).

What are the possible limits Y ? Properties of Y (µ)?

Finite energy measures

M1: signed measures of finite total variation |µ|.

For α 6= d define subspace of measures

Mα ={µ ∈M1;

∫Rd×Rd

|z − z′|d−α|µ|(dz)|µ|(dz′) < +∞ and

∫Rd×Rd

|z − z′|2kµ(dz)µ(dz′) = 0 all k s.t. 0 ≤ 2k < d− α}

For β ∈ (0, 2d) with β 6= d, if α > β > d or if α < β < d thenMα ⊂Mβ . Define

Mβ =

⋃

α∈(β,2d)

Mα if β > d

⋃α∈(0,β)

Mα if β < d

Remarks:

I Mα = {0}, if α ≥ 2d

I Mα ={µ ∈M1;

∫Rd×Rd |z − z′|d−α|µ|(dz)|µ|(dz′) < +∞

}, if

d < α < 2d

I Condition∫Rd×Rd |z − z′|2kµ(dz)µ(dz′) = 0 can be replaced by∫

Rd

zj11 z

j22 . . . zjd

d µ(dz) = 0

for integers j1, j2, . . . , jd such that 0 ≤ j1 + j2 + . . .+ jd < d− α

(c.f Matheron 1973, where similar spaces of measures are introduced)

Fractional Gaussian limit

Thm: Let F (dr) ∼ Cβr−β−1 be as above with β < 2d such that

β − d 6∈ 2N. If λρβ →∞ as ρ→ 0β−d, then

Yλ,ρ(µ)√Cβλρβ

f.d.d.=⇒ constWβ(µ), µ ∈ Mβ

where Wβ is a Gaussian random field µ→Wβ(µ) with covariancefunctional

E (Wβ(µ)Wβ(ν)) = cβ

∫Rd

∫Rd

|z − z′|d−β µ(dz)ν(dz′),

and the constant cβ ensures that the covariance is of positive type.

Intermediate limit

Thm: Again, take F (dr) ∼ Cβr−β−1 with β < 2d such that β − d 6∈ 2N.

For arbitrary c > 0, if

λρβ → cβ−d as ρ→ 0β−d,

then

Yλ,ρ(µ)f.d.d.=⇒ constYβ(µc),

where µc(A) = µ(cA), A ⊂ Rd, µ ∈ Mβ , and

Yβ(µ) =∫Rd

∫ ∞

0

µ(B(x, r)) (Nβ(dx, dr)− Cβ dx r−β−1dr).

Covariance

In heavy-tailed case d < β < 2d we have

Var(Wβ(µ)) = Var(Yβ(µ)) = const∫Rd

∫Rd

µ(dy)µ(dy′)|y − y′|β−d

.

For microballs case d− 1 < β < d (0 < H < 1/2), the covariancefunctional has the representation

E (Wβ(µ)Wβ(ν))

=12

∫Rd

∫Rd

(|z|d−β + |z′|d−β − |z − z′|d−β

)µ(dz)ν(dz′).

Similarity properties

I the Gaussian field is self-similar with indexH = (d− β)/2 ∈ (−d/2,∞), i.e. for all s > 0,

W (µs)fdd= sHW (µ), where µs(A) = µ(A/s).

I the intermediate limit is “aggregate-similar” in the sense thatthe scaled process passes through all aggregates of i.i.d.copies: there exists a sequence cm →∞ such that

Yβ(µ1/cm)

fdd=

m∑i=1

Y iβ(µ)

Case d = 1

If 1 < β < 2 then H = (1− β)/2 ∈ (−1/2, 0) and

Var(Wβ(µ)) =∫R

∫R

µ(du)µ(dv)|u− v|−2H

.

The choice µt(du) = 1[0,t](u) du is in Mβ , and yields

Var(Wβ(µt)) =∫ t

0

∫ t

0

du dv

|u− v|2(1−H′)= const t2H′

,

with H ′ = H + 1, so BH(t) = Wβ(µt) is ordinary FBM with1/2 < H ′ < 1.

For 0 < β < 1, take

µt(du) = δt(du)− δ0(du) ∈ Mβ .

Then BH(t) = Wβ(µt) is ordinary FBM with 0 < H < 1/2.

Fractional Poisson motion

Consider intermediate limit in Rd, microballs case d− 1 < β < d. Takeµ = δx − δ0, which belongs to Mβ . Then

x→ Yβ(x) = Yβ(δx − δ0)

has a symmetrized Poisson distribution with mean |x|2H and covariance

Cov(Yβ(x), Yβ(y)) =12(|x|2H+|y|2H−|x−y|2H), H =

β − d2

∈ (0, 1/2)

Comparison with short range case

Thm: Assume ER2d <∞. As ρ→ 0, λ→∞,

Yλ,ρ(µ)(λρ2d)1/2E(R2d)1/2

f.d.d=⇒ W (µ), µ(dx) = φ(x) dx, φ ∈ L1 ∩ L2

where W (µ) is the Gaussian functional with covariance∫Rd φ(x)ψ(x) dx (white noise).