A Gaussian/Poisson alternative on configuration spaces Giovanni - - PowerPoint PPT Presentation

A Gaussian/Poisson alternative on configuration spaces

Giovanni Peccati (Luxembourg University)

SSP 2012/Kansas University: March 24, 2012

Giovanni Peccati Gaussian/Poisson

Introduction

Partially based on joint works with R. Lachièze-Rey (Paris V). This is one of the latest installments in a recent series of papers, focussing on probabilistic approximations by means of the Malliavin calculus of variations and the Stein’s and Chen-Stein method for probabilistic approximations. Strong motivations come from stochastic geometry, in particular in connection with the notion of geometric U-statistics (Reitzner and Schulte, 2011).

Giovanni Peccati Gaussian/Poisson

An example

Let η be a Poisson measure on R2, with control equal to the Lebesgue measure. Define Wn =

• −1

2 √ n, 1 2 √ n

2

, n = 1, 2, ..., . Let {rn} be a non-increasing sequence of positive

• numbers. For every n, we consider the disk graph

Gn = (Vn, En), where Vn = Wn ∩ η, En = {(x, y) : 0 < |x − y| < rn}. We are interested in the asymptotic behavior of Mn = #{edges of Gn},

• Mn = Mn − E[Mn]
• Var(Mn) .

Giovanni Peccati Gaussian/Poisson

An example

One has that: (i) If nr 2

n → ∞,

then Mn

LAW

→ N(0, 1); (ii) If nr 2

n → c ∈ (0, +∞),

then Mn

TV

→ Poisson; (iii) If nr 2

n → 0,

then Mn, Mn

L1

→ 0. Remark: Same array of behaviors in any dimension, and with different graphical rules (P . - Lachièze-Rey, 2011).

Giovanni Peccati Gaussian/Poisson

General aim

We shall develop unified analytic techniques ( = based on differential operators) allowing to: (a) Measure the quantity dW(F, N), for F a regular functional

• f a Poisson measure and N Gaussian;

(b) Measure the quantity dTV(F, Po(λ)), for F a Z+-valued functional of a Poisson measure.

Giovanni Peccati Gaussian/Poisson

Related research

Similar results in a Gaussian framework: several papers since 2005, e.g. by I. Nourdin, D. Nualart, G.P ., ... (forthcoming book, 2012). Applications: Breuer-Major Theorems, harmonic analysis, density estimates, concentration inequalities, needlet estimation, ... Stochastic geometry: Reitzner and Schulte (2011), Schulte and Thaele (2011, 2012), Decreusefond et al. (2011). A U-statistic is an object of the type: F =

• {x1,...,xd}∈ηd

=

ϕ(x1, ..., xd). Very few (extremely technical) results about CLTs for non-Poissonized U-statistics: de Jong (1990), Jammalamadaka and Janson (1986), Battahkhraya and Ghosh (1992).

Giovanni Peccati Gaussian/Poisson

Digression: a connection with universality

Let {ξi : i ≥ 1} be i.i.d. N(0, 1) and fix d ≥ 2. In Nourdin, P ., Reinert (2010), it is proved that if

• a(n)

i1,...,idξi1 · · · ξid LAW

→ N(0, 1), then a(n)

i1,...,idYi1 · · · Yid verifies the same CLT for every

centered i.i.d. sequence {Yi : i ≥ 1} with unit variance. This result is very much in line with recent examples of “universality of discrete Gaussian families”: see e.g. recent proofs of the circular law, or generalized Lindberg principles by Rotar’ (1979), Mossel, O’Donnel and Oleszkiewicz (2008) or Chatterjee (2007).

Giovanni Peccati Gaussian/Poisson

Digression: a connection with universality

Let G be a Gaussian measure with Lebesgue control. Our results show that there exist multilinear functionals Φn s.t. Φn(G) verifies a CLT, but Φn( η) is asymptotically Poisson. This shows that a naive version of the universality phenomenon for random measures fails.

Giovanni Peccati Gaussian/Poisson

Poisson measures

(Z, Z) is a Polish space. Given a σ-finite non atomic measure µ, we denote by η a Poisson measure with control µ, and its compensated counterpart is η = η(·) − µ(·). Recall: for every A, B such that A ∩ B = ∅ and µ(A), µ(B) < ∞, η(A) and η(B) are two independent Poisson r.v.’s of parameters µ(A), µ(B).

Giovanni Peccati Gaussian/Poisson

Integral and chaos

For every symmetric square-integrable function f in q variables, we define the multiple Wiener-Itô integral Iq(f) =

• Z

· · ·

• Z

f(x1, ..., xq)1{no diagonals}ˆ η(dx1) · · · ˆ η(dxq). Recall that every F ∈ L2(σ(η)) can be written as: F = E(F) + ∞

q=1 Iq(fq).

Giovanni Peccati Gaussian/Poisson

A tiny bit of Malliavin calculus

The derivative operator is: DzF =

q qIq−1(fq(z, ·)).

Nualart and Vives (1990): DzF(η) = F(η + δz) − F(η) (add-one cost). The O-U generator: LF = −

q≥1 qIq(fq).

Pseudo-inverse of the O-U generator: L−1F = −

q≥1 q−1Iq(fq).

Integration by parts: for every X derivable and F centered, E[XF] = E[DX, −DL−1Fµ].

Giovanni Peccati Gaussian/Poisson

Another look at U-statistics

U-statistics are typically “smooth functionals”. For instance, using η = ˆ η + µ: F =

• {x1,x2}∈ηd

=

ϕ(x1, x2) = ϕ(x, y)1{x=y}η(dx)η(dy) = E[F] + 2I1(f) + I2(ϕ), where f(x) =

ϕ(x, y)µ(dy).

Giovanni Peccati Gaussian/Poisson

Stein’s Lemma and equations

(Stein’s Lemma) A random variable Z has the N(0, 1) distribution if and only if E[Zf(Z) − f ′(Z)] = 0 for every smooth function f. (Stein’s equations) Let Z ∼ N(0, 1). For every h ∈ Lip(1), the equation f ′(x) − xf(x) = h(x) − E[h(Z)] admits a solution fh such that f ′

h∞ ≤ 1 and f ′′ h ∞ ≤ 2.

Giovanni Peccati Gaussian/Poisson

Applying integration by parts

Now consider F centered and differentiable: for every f smooth E[Ff(F)] = E[Df(F), −DL−1Fµ] = E[f ′(F)DF, −DL−1Fµ]+Rf, where |Rf| ≤ 1

2f ′′∞E

• Z(DzF)2|DzL−1F|µ(dz). Then, for

Y ∼ N(0, 1), dW(F, Y) ≤ sup

|f ′|≤1,|f ′′|≤2

• E[f ′(F)] − E[Ff(F)]
• ≤

sup

|f ′|≤1,|f ′′|≤2

• E[f ′(F)(1 − DF, −DL−1Fµ] + Rf
• ≤ E|1 − DF, −DL−1Fµ| + E
• Z

(DzF)2|DzL−1F|µ(dz).

Giovanni Peccati Gaussian/Poisson

First bound

Theorem 1 (P ., Solé Taqqu and Utzet, 2010) Let Z ∼ N(0, 1). For every differentiable centered F ∈ L2(σ(η)) dW(F, Z) ≤ E|1 − DF, −DL−1Fµ| +E

• Z

(DzF)2|DzL−1F|µ(dz). If F = Iq(f), dW(F, Z) ≤ E

• 1 − 1

q µ(|DF|2)

• + 1

q Eµ(|DF|3).

Giovanni Peccati Gaussian/Poisson

Chen-Stein’s Lemma and equations

(Chen-Stein Lemma) A random variable Y in Z+ has the Po(λ) distribution if and only if E[Yg(Y) − λg(Y + 1)] = 0 for every bounded function g. (Chen-Stein equations) Let Y ∼ Po(λ). For every h = 1A, the equation λg(k + 1) − kg(k) = h(k) − E[h(Y)], k = 0, 1, 2, ..., admits a unique bounded solution gA such that gA(0) = 0, ∆gA∞ ≤ 1−e−λ

λ

and ∆2gA∞ ≤ 2λ−1∆gA∞.

Giovanni Peccati Gaussian/Poisson

Applying integration by parts

Recall that, for g : Z+ → R bounded, |g(a) − g(k) − ∆g(k)(a − k)| ≤ 1 2∆2g∞|(a − k)(a − k − 1)|. Now consider F with values in Z+, differentiable and such that E[F] = λ: for every g bounded E[Fg(F) − λg(F + 1)] = E[(F − λ)g(F) − λ∆g(F)] = E[Dg(F), −DL−1Fµ − λ∆g(F)] = E[∆g(F)(DF, −DL−1Fµ − λ)] + Rg, where |Rg| ≤ 1

2∆2g∞E

• Z |DzF(DzF − 1)DzL−1F|µ(dz).

Giovanni Peccati Gaussian/Poisson

Applying integration by parts

Then, for Y ∼ Po(λ), and writing Bλ = 1−e−λ

λ

, Cλ = 1−e−λ

λ2

. dTV(F, Y) = sup

A

|E[FgA(F)] − λE[gA(F + 1)]| ≤ sup

A

• E[∆gA(F)(DF, −DL−1Fµ − λ)] + RgA
• ≤ BλE|λ − DF, −DL−1Fµ|

+CλE

• Z

|(DzF)(DzF − 1)DzL−1F|µ(dz).

Giovanni Peccati Gaussian/Poisson

Second bound

Theorem 2 (P ., 2011) Let Y ∼ Po(λ). For every differentiable F ∈ L2(σ(η)) with values in Z+ and such that E[F] = λ, dTV(F, Y) ≤ BλE|λ − DF, −DL−1Fµ| +CλE

• Z

|(DzF)(DzF − 1)DzL−1F|µ(dz). If F = λ + Iq(f), dTV(F, Y) ≤ BλE

• λ − 1

q µ(|DF|2)

• + Cλ

q Eµ(|(DF)2(DF − 1)|). Remark: E[DF, −DL−1Fµ] = Var(F).

Giovanni Peccati Gaussian/Poisson

To resume

For a smooth centered random variable F, dW(F, N) ≤ E|1 − DF, −DL−1Fµ| +E

• Z

(DzF)2|DzL−1F|µ(dz). For a Z+-valued random variable F of mean λ, dTV(F, Po(λ)) ≤ BλE|λ − DF, −DL−1Fµ| +CλE

• Z

|(DzF)(DzF − 1)DzL−1F|µ(dz).

• Remark. On a Gaussian space (Nourdin, P

. 2009): dTV(F, N) ≤ 2E|1 − DF, −DL−1Fµ|.

Giovanni Peccati Gaussian/Poisson

Back to U-statistics

In the disk graph model, whenever rn → 0, one can show that Mn = {number of edges in Gn} = E[Mn] + I2(ϕn) + negligible terms, where ϕn(x, y) = 1

210<|x−y|<rn. Moreover, one has that

E[Mn] → ∞ or some λ > 0, according as nr 2

n tends to ∞ or to

some constant, since nr 2

n ≍ E[Mn] ∼ Var(Mn).

Giovanni Peccati Gaussian/Poisson

Bounds for integrals

Let F = λ + I2(f) and σ2 = Var(F). The two bounds computed before become dW( F, N(0, 1)) ≤ C1 σ2 × {f ⋆1

1 f + f ⋆1 2 f + µ(f 4)1/2}

dTV(F, Po(λ′)) ≤ |λ − λ′| +C2 × {f ⋆1

1 f + f ⋆1 2 f + µ(4f 2 + 16f 4 − 16f 3)1/2},

where f ⋆1

1 f(x, y) =

f(x, a)f(y, a)µ(da) and

f ⋆1

2 f(x) =

f 2(x, a)µ(da).

Giovanni Peccati Gaussian/Poisson

Bounds for integrals

In particular, when f = 1

21H,

dW( F, N(0, 1)) ≤ C1 σ2 × {f ⋆1

1 f + f ⋆1 2 f} + C1

σ dTV(F, Po(λ′)) ≤ |λ − λ′| + C2 × {f ⋆1

1 f + f ⋆1 2 f}.

When applied to our example, dW( Mn, N(0, 1)) ≤ C1 rn √n dTV(Mn, Po(λ′)) ≤ |nr 2

n − λ′| + C2rn.

Giovanni Peccati Gaussian/Poisson

A more general picture

Assume now that x, y ∈ Wn are connected if and only if x − y ∈ Hn, where Hn is some symmetric (possibly infinite) set. We consider the occupation coefficients ψ(n) = Leb(Hn ∩ Wn) Leb(Wn) , n ≥ 1. Then (P .-Lachièze-Rey, 2011), (i) If n2ψ(n) → ∞, Mn

W

→ N(0, 1) (dW ≤ C max{n−1/2, (n2ψ(n))−1/2}); (ii) If n2ψ(n) → c ∈ (0, +∞), Mn

TV

→ Poisson (dTV ≤ C(nψ(n))1/2}); (iii) If n2ψ(n) → 0, Mn, Mn

L1

→ 0.

Giovanni Peccati Gaussian/Poisson

Remarks

The bounds hold for U-statistics and multiple integrals of arbitrary orders. Sufficient conditions for central and Poisson limit theorems expressed in terms of contractions: much simpler than those obtained by de-Poissonization. These conditions turn out to be necessary and sufficient in a number of instances, as well as to be equivalent to conditions that can be expressed in terms of moments. Extensions to more complex models: like e.g. the Boolean model (P . Lachièze-Rey, 2012).

Giovanni Peccati Gaussian/Poisson

Some further directions

Multidimensional Poisson Clumping (compound Poisson) Functional versions (even in the Gaussian framework) Comparisons with alternate definitions of Malliavin

• perators; connections with a paper by Ledoux.

Upper bounds in geometric models with infinite chaotic expansions Optimality of bounds Combinatorial interpretations (relations with factorial cumulants) Charlier-Edgeworth expansions

Giovanni Peccati Gaussian/Poisson