Steins method for normal approximation of linear statistics of - - PowerPoint PPT Presentation

Stein’s method for normal approximation of linear statistics of beta-ensembles

Gaultier Lambert (joint work with Michel Ledoux and Christian Webb)

UZH University of Z¨ urich 1

Stein Method

2

Normal approximation for Gibbs measures

3

β-ensembles

Stein Method

Stein Method

When is a random variable Gaussian?

Lemma (Stein ’72) Let X be a real-valued random variable. X is N(0, σ2) if and only if for all f ∈ C ∞

b (R),

E

• Xf (X) − σ2f ′(X)
• = 0.

Stein Method

When is a random variable Gaussian?

Lemma (Stein ’72) Let X be a real-valued random variable. X is N(0, σ2) if and only if for all f ∈ C ∞

b (R),

E

• Xf (X) − σ2f ′(X)
• = 0.

Let (Xn)n∈N be a sequence of random variables. If for all f ∈ C ∞

b (R),

lim

n→∞ E

• Xnf (Xn) − f ′(Xn)
• = 0,

then Xn ⇒ N(0, 1).

Stein Method

Proof:

Let Z be N(0, 1). It suffices to show that for any ϕ ∈ C ∞(R), 0 ≤ ϕ ≤ 1, lim

n→∞ Eϕ(Xn) = Eϕ(Z).

Stein Method

Proof:

Let Z be N(0, 1). It suffices to show that for any ϕ ∈ C ∞(R), 0 ≤ ϕ ≤ 1, lim

n→∞ Eϕ(Xn) = Eϕ(Z).

Consider the equation: f ′(x) = xf (x) − ϕ(x) + Eϕ(Z). The solution is given by f (x) = −ex2/2 ˆ x

−∞

e−t2/2 ϕ(t) − Eϕ(Z)

• dt

= ex2/2 ˆ ∞

x

e−t2/2 ϕ(t) − Eϕ(Z)

• dt

Stein Method

Proof:

Let Z be N(0, 1). It suffices to show that for any ϕ ∈ C ∞(R), 0 ≤ ϕ ≤ 1, lim

n→∞ Eϕ(Xn) = Eϕ(Z).

Consider the equation: f ′(x) = xf (x) − ϕ(x) + Eϕ(Z). The solution is given by f (x) = −ex2/2 ˆ x

−∞

e−t2/2 ϕ(t) − Eϕ(Z)

• dt

= ex2/2 ˆ ∞

x

e−t2/2 ϕ(t) − Eϕ(Z)

• dt

Thus, we have Eϕ(Xn) = Eϕ(Z) + E

• Xnf (Xn) − f ′(Xn)
• and, since f is smooth and bounded by

√ 2πϕL∞, the last term converges to zero as n → ∞.

Stein Method

Probability metrics

If X is a complete separable metric space, the space of Borel probability measures on X equipped with the topology of weak convergence is completely metrizable. An example of a metric is given by d(X, Y ) = sup

ϕ∈F

• Eϕ(X) − Eϕ(Y )
• where F is a sufficiently big class of functions ϕ : X → R.

Stein Method

Probability metrics

If X is a complete separable metric space, the space of Borel probability measures on X equipped with the topology of weak convergence is completely metrizable. An example of a metric is given by d(X, Y ) = sup

ϕ∈F

• Eϕ(X) − Eϕ(Y )
• where F is a sufficiently big class of functions ϕ : X → R.

The total variation distance: dTV(X, Y ) = sup

ϕL∞ ≤1/2

• Eϕ(X) − Eϕ(Y )
• .

The Kolmogorov distance (for real-valued random variables): dK(X, Y ) = sup

x∈R

• P[X ≤ x] − P[Y ≤ x]
• .

Stein Method

Kantorovich or Wasserstein metrics

For any q ≥ 1, define the Wasserstein q distance between two random variables: Wq(X, Y ) = inf

(x,y) E [|x − y|q]1/q .

where the infimum is taken over all couplings or random vectors (x, y) such that x

d

= X and y

d

= Y .

Stein Method

Kantorovich or Wasserstein metrics

For any q ≥ 1, define the Wasserstein q distance between two random variables: Wq(X, Y ) = inf

(x,y) E [|x − y|q]1/q .

where the infimum is taken over all couplings or random vectors (x, y) such that x

d

= X and y

d

= Y . It turns out that W1(X, Y ) = sup

ϕLip≤1

• Eϕ(X) − Eϕ(Y )
• .

Stein Method

Kantorovich or Wasserstein metrics

For any q ≥ 1, define the Wasserstein q distance between two random variables: Wq(X, Y ) = inf

(x,y) E [|x − y|q]1/q .

where the infimum is taken over all couplings or random vectors (x, y) such that x

d

= X and y

d

= Y . It turns out that W1(X, Y ) = sup

ϕLip≤1

• Eϕ(X) − Eϕ(Y )
• .

The bounded-Lipschitz distance: dBL(X, Y ) = sup

ϕLip≤1 ϕL∞ ≤1

• Eϕ(X) − Eϕ(Y )
• .

Stein Method

Berry-Essen theorem

Let (Xj)j∈N be i.i.d. real-valued random variables with mean zero and variance 1 and define SN = X1 + · · · + XN √ N . Theorem (Barbour, Hall, ’84) There exists C > 0 such that dK(SN, Z) = sup

x∈R

• E[SN ≤ x] − E[Z ≤ x]
• ≤ C E|X1|3

√ N .

Stein Method

Berry-Essen theorem

Let (Xj)j∈N be i.i.d. real-valued random variables with mean zero and variance 1 and define SN = X1 + · · · + XN √ N . Theorem (Barbour, Hall, ’84) There exists C > 0 such that dK(SN, Z) = sup

x∈R

• E[SN ≤ x] − E[Z ≤ x]
• ≤ C E|X1|3

√ N . We will now show the easier claim: dBL(SN, Z) ≤ C E|X1|3 √ N .

Stein Method

Regularity of the solution of Stein’s equation

Stein’s Lemma The solution of Stein’s equation satisfy f L∞ ≤ √ 2πϕL∞, f ′L∞ ≤ min{ϕLip, 2ϕL∞} f ′Lip ≤ 4ϕL∞ + 2ϕLip

Stein Method

Proof:

For all t ∈ [0, 1], let St

N = tX1 + X2 + · · · + XN

√ N . Let ϕ ∈ L∞(R), Lipschitz–continuous, and f be the corresponding solution of Stein’s

• equation. By Taylor’s theorem, we have

E [SNf (SN)] = √ NE

• X1f (S1

N)

• =

√ NE

• X1
• f (S0

N) + f ′(ST N )X1/

√ N

• =

√ NE

• X1f (S0

N)

• + E
• X 2

1 f ′(ST N )

• where T is uniform in [0, 1] and independent of (Xj)j∈N.

Stein Method

Proof:

For all t ∈ [0, 1], let St

N = tX1 + X2 + · · · + XN

√ N . Let ϕ ∈ L∞(R), Lipschitz–continuous, and f be the corresponding solution of Stein’s

• equation. By Taylor’s theorem, we have

E [SNf (SN)] = √ NE

• X1f (S1

N)

• =

√ NE

• X1
• f (S0

N) + f ′(ST N )X1/

√ N

• =

√ NE

• X1f (S0

N)

• + E
• X 2

1 f ′(ST N )

• where T is uniform in [0, 1] and independent of (Xj)j∈N.

By independence, E

• X1f (S0

N)

• = 0

and E

• X 2

1 f ′(S0 N)

• = E
• f ′(S0

N)

• ,

so that E [SNf (SN)] = E

• X 2

1

• f ′(ST

N ) − f ′(S0 N)

• + E
• f ′(S0

N)

• .

Stein Method

Proof:

Thus, we have E

• SNf (SN) − f ′(SN)
• = E
• X 2

1

• f ′(ST

N ) − f ′(S0 N)

• + E
• f ′(S0

N) − f ′(S1 N)

• Since f ′ is Lipschitz–continuous and ST

N − S0 N = TX1 √ N , we obtain

E

• SNf (SN) − f ′(SN)
• ≤ 3f ′Lip

2 √ N E

• |X1|3

.

Stein Method

Proof:

Thus, we have E

• SNf (SN) − f ′(SN)
• = E
• X 2

1

• f ′(ST

N ) − f ′(S0 N)

• + E
• f ′(S0

N) − f ′(S1 N)

• Since f ′ is Lipschitz–continuous and ST

N − S0 N = TX1 √ N , we obtain

E

• SNf (SN) − f ′(SN)
• ≤ 3f ′Lip

2 √ N E

• |X1|3

. By Stein’s equation: f ′(x) = xf (x) − ϕ(x) + Eϕ(Z), we conclude that, if ϕLip ≤ 1 and ϕL∞ ≤ 1, then E [ϕ(SN) − ϕ(Z)] ≤ 9 √ N E

• |X1|3

. so that dBL(SN, Z) ≤ 9E

• |X1|3

√ N .

Stein Method

Theorem (Chatterjee, Meckes ’08) Let (Xt)t≥0 be a family of random variables in L2(P) with the same law. Assume that X0 is F–measurable and that there exists constants Λ, σ > 0 such that lim

t→0 E

Xt − X0 t

• F
• = −ΛX0 + R

lim

t→0 E

(Xt − X0)2 t

• F
• = 2Λσ2 + S

E

• |Xt − X0|2+α

= o(t)

t→0

, α > 0. Then W1(X0, σZ) ≤ E|R| + E|S| Λ .

Stein Method

Theorem (Chatterjee, Meckes ’08) Let (Xt)t≥0 be a family of random variables in L2(P) with the same law. Assume that X0 is F–measurable and that there exists constants Λ, σ > 0 such that lim

t→0 E

Xt − X0 t

• F
• = −ΛX0 + R

lim

t→0 E

(Xt − X0)2 t

• F
• = 2Λσ2 + S

E

• |Xt − X0|2+α

= o(t)

t→0

, α > 0. Then W1(X0, σZ) ≤ E|R| + E|S| Λ . Applications in random matrix theory: Chatterjee ’09 D¨

• bler–Stolz ’11 ’14

Fulman ’12 Webb ’16 Joyner–Smilansky ’15 ’17

Stein Method

Proof:

Let ϕ be a Lipschitz continuous and consider the solution of the equation: (1) h′′(x) − xh′(x) = ϕ(x) − Eϕ(Z). Since Xt and X0 have the same law, for any t > 0, E [h(Xt) − h(X)] = 0. By Taylor’s theorem, we have E

• (Xt − X0)h′(X0) + (Xt − X0)2

2 h′′(X0)

• = O

t→0

• E|Xt − X0|2+α

. By assumption (σ2 = 1), we also have lim

t→0 E

(Xt − X0) t h′(X0) + (Xt − X0)2 2t h′′(X0)

• = 0

= ΛE

• −X0h′(X0) + σ2h′′(X0)
• + E
• Rh′(X0) + Sh′′(X0)/2
• .

Using equation (1), since h′L∞, h′′L∞ ≤ ϕLip, we obtain Eϕ(X0) − Eϕ(Z) ≤ E|R| + E|S| Λ .

Normal approximation for Gibbs measures

Normal approximation for Gibbs measures

Gibbs measure

Let P(dx) = e−W (x)dx be a probability measure on RN. If W is a sufficiently nice function, there exists a Markov diffusion Zt with invariant measure P: dZt = −∇W (Zt)dt + √ 2dBt.

Normal approximation for Gibbs measures

Gibbs measure

Let P(dx) = e−W (x)dx be a probability measure on RN. If W is a sufficiently nice function, there exists a Markov diffusion Zt with invariant measure P: dZt = −∇W (Zt)dt + √ 2dBt. Formally, by Ito’s formula, the generator is given by L f (x) := d dt EZ0=x

• f (Zt)
• t=0

= −∇W (x) · ∇f (x) + ∆f (x). Moreover, if f , g are real-valued function in the domain of L , then − ˆ g(x)L f (x) P(dx) = ˆ ∇g(x) · ∇f (x) P(dx).

Normal approximation for Gibbs measures

Approximate eigenfunction property

Let φ : RN → R be smooth and Xt = φ(Zt) where Z0 = λ is a random configuration with law P. Assume that there exists Λ > 0 such that L φ = −Λφ + R and

• R(λ)
• ≪ Λ. Then, we have

lim

t→0 E

Xt − X0 t |λ

• = −Λφ(λ) + R(λ)

lim

t→0 E

(Xt − X0)2 t |λ

• = 2|∇φ(λ)|2

lim

t→0 E

|Xt − X0|2+α t

• = 0.

Normal approximation for Gibbs measures

Approximate eigenfunction property

Let φ : RN → R be smooth and Xt = φ(Zt) where Z0 = λ is a random configuration with law P. Assume that there exists Λ > 0 such that L φ = −Λφ + R and

• R(λ)
• ≪ Λ. Then, we have

lim

t→0 E

Xt − X0 t |λ

• = −Λφ(λ) + R(λ)

lim

t→0 E

(Xt − X0)2 t |λ

• = 2|∇φ(λ)|2

lim

t→0 E

|Xt − X0|2+α t

• = 0.

Moreover, we expect that when N is large, |∇φ(λ)|2 = E|∇φ|2 + S(λ). This implies that W1

• φ(λ), σZ
• ≤ E|R| + E|S|

Λ with σ2 = E|∇φ|2 Λ .

Normal approximation for Gibbs measures

Multidimensional Normal approximation result

Let Φ : RN → Rd, Λ = diag(κ1, · · · , κd) and Σ = diag(σ1, · · · , σd), where κj, σj > 0. Theorem Let λ be random configuration with law P and suppose that Φ, ∇Φ ∈ L2(P), then W2

• Φ(λ), Nd(0, Σ2)
• ≤ AL2(P) + BL2(P).

where A = Φ + Λ−1L Φ, B = Λ−1(∇Φ)(∇Φ)T − Σ2.

Normal approximation for Gibbs measures

Normal approximation for general test functions

Let Φn : RN → R be a family of functions such that EΦn = 0, ΦnL2(P) ≤ Nǫ, and W2

• Φn(λ)

dN

n=1, NdN (0, Σ2)

• ≤ δN

for some sequences dN → ∞ and δN → 0 as N → ∞.

Normal approximation for Gibbs measures

Normal approximation for general test functions

Let Φn : RN → R be a family of functions such that EΦn = 0, ΦnL2(P) ≤ Nǫ, and W2

• Φn(λ)

dN

n=1, NdN (0, Σ2)

• ≤ δN

for some sequences dN → ∞ and δN → 0 as N → ∞. Suppose that f = ∞

n=1

fnΦn and Cf =

∞

• n=1

| fn| < ∞, σ2

f = ∞

• n=1
• f 2

n σ2 n < ∞.

Proposition We have W2

• f (λ), σf Z
• ≤ Nǫ

n>dN

| fn| +

• n>dN
• f 2

n σ2 n + C 2 f δN.

β-ensembles

β-ensembles

Definition

Let V ∈ C 2(R) such that infx∈R V ′′(x) > −∞ and there exists c > 1, V (x) ≥ c log |x| as |x| → ∞. Let W (x) =

• 1≤i<j≤N

log |xi − xj|−1 + N

N

• i=1

V (xi) and consider the probability measure PN

V ,β on RN < with density

1 Z N

V ,β

e−βW (x), β > 0.

β-ensembles

Gaussian ensembles

If β = 2 and V (x) = x2, this describes the joint law of the eigenvalues of the Gaussian Unitary Ensemble (GUE):      Mii

d

= NR(0,

1 2N )

1 ≤ i ≤ N Mij

d

= NR(0,

1 2N )

1 ≤ i < j ≤ N Mji = Mij 1 ≤ i < j ≤ N .

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

Figure: Histogram of the eigenvalues of a 2000 × 2000 GUE matrix. — the density of the semicircular law, µsc(dx) = 2

π

√ 1 − x21|x|<1dx.

β-ensembles

Equilibrium measure

We consider the empirical measure µN(dx) = 1 N

• 1≤j≤N

δλj , λ = (λ1, . . . , λN) has law PN

V ,β

There exists a probability measure µV on R with compact support such that for any g ∈ C (R) bounded, almost surely, lim

N→∞

ˆ g dµN = ˆ g dµV .

β-ensembles

Equilibrium measure

We consider the empirical measure µN(dx) = 1 N

• 1≤j≤N

δλj , λ = (λ1, . . . , λN) has law PN

V ,β

There exists a probability measure µV on R with compact support such that for any g ∈ C (R) bounded, almost surely, lim

N→∞

ˆ g dµN = ˆ g dµV . For instance, in the Gaussian case, V (x) = x2, the equilibrium measure is the Wigner semicircle law: µsc(dx) = 2 π

• 1 − x21|x|<1dx.

β-ensembles

One-cut assumption

We assume that supp(µV ) = [−1, 1]. Then, the equilibrium measure satisfies the condition ˆ 1 x − t µV (dt) = V ′(x), ∀x ∈ [−1, 1], and µV (dx) = S(x)µsc(dx). We also assume that S(x) > 0 for all x ∈ [−1, 1].

β-ensembles

Mean field operator

Define the operator Ξx(ψ) = −V ′(x)ψ(x) + ˆ ψ(x) − ψ(t) x − t µV (dt). If g ∈ C k(R), then the equation Ξ(ψ) = g − cg, cg = ˆ 1

−1

g(x) dx π √ 1 − x2 has a unique solution ψ ∈ C k−1([−1 − ǫV , 1 + ǫV ]) for some ǫV > 0.

β-ensembles

Mean field operator

Define the operator Ξx(ψ) = −V ′(x)ψ(x) + ˆ ψ(x) − ψ(t) x − t µV (dt). If g ∈ C k(R), then the equation Ξ(ψ) = g − cg, cg = ˆ 1

−1

g(x) dx π √ 1 − x2 has a unique solution ψ ∈ C k−1([−1 − ǫV , 1 + ǫV ]) for some ǫV > 0. We define m(g) = − ˆ ψ′ dµV and Σ(g) = −1 2 ˆ g ′ψ dµV . It turns out that Σ does not depend on the potential V and Σ(g) = 1 4

• k≥1

kg 2

k ,

gk = 2 ˆ 1

−1

g(x)Tk(x) dx π √ 1 − x2 , where (Tk)k>1 denote the Chebyshev’s polynomials of the first kind: Tk(cos θ) = cos(kθ).

β-ensembles

Fluctuations

We define the random measure νN = N(µN − µV ) so that ˆ g dνN :=

N

• j=1

g(λj) − N ˆ g dµV . Main result Suppose that V ∈ C 8(R) satisfies the one-cut condition and g ∈ C 9(R) with at most polynomial growth, then lim

N→∞ W2

ˆ g dνN, Z

• = 0

where Z

d

= N

• 1

2 − 1 β

• m(g), 2

β Σ(g)

• .

Previous work on fluctuations of β-ensembles: Johansson ’98 Cabanal-Duvillard ’01 Shcherbina, M. ’10 ’13 ’14 Borot–Guionnet ’13 Bekerman–Lebl´ e–Serfaty ’17

β-ensembles

Further results

Theorem Suppose that V ∈ C ∞(R) satisfies the one-cut condition and g ∈ C ∞

c

• (−1, 1)
• ,

then for any ǫ > 0, W2 ˆ g dνN, Z

• ≤ Nǫ−1,

when N is sufficiently large.

β-ensembles

Further results

Theorem Suppose that V ∈ C ∞(R) satisfies the one-cut condition and g ∈ C ∞

c

• (−1, 1)
• ,

then for any ǫ > 0, W2 ˆ g dνN, Z

• ≤ Nǫ−1,

when N is sufficiently large. Theorem In the GUE case, V (x) = x2 and β = 2, for any real-valued polynomial Q, we have W2 ˆ Q νN, Z

• ≤ CQ

N where Z

d

= N (0, Σ(Q)) .

β-ensembles

Dyson’s Brownian motion

Let λ be random configuration with law PN

V ,β(dx) =

1 Z N

V ,β

e−βW (x)dx, W (x) =

• 1≤i<j≤N

log |xi − xj|−1 + N

N

• i=1

V (xi). The infinitesimal generator is given by L = ∆ − β∇W · ∇.

β-ensembles

Dyson’s Brownian motion

Let λ be random configuration with law PN

V ,β(dx) =

1 Z N

V ,β

e−βW (x)dx, W (x) =

• 1≤i<j≤N

log |xi − xj|−1 + N

N

• i=1

V (xi). The infinitesimal generator is given by L = ∆ − β∇W · ∇. If f (λ) = N

n=1 φ(λj) =

ˆ φ dνN + N ˆ φ dµV , then L f (λ) =

N

• j=1

φ′′(λj) − βN

N

• j=1

V ′(λj)φ′(λj) + β

N

• i,j=1

i=j

φ′(λi) λi − λj

β-ensembles

Dyson’s Brownian motion

Let λ be random configuration with law PN

V ,β(dx) =

1 Z N

V ,β

e−βW (x)dx, W (x) =

• 1≤i<j≤N

log |xi − xj|−1 + N

N

• i=1

V (xi). The infinitesimal generator is given by L = ∆ − β∇W · ∇. If f (λ) = N

n=1 φ(λj) =

ˆ φ dνN + N ˆ φ dµV , then L f (λ) =

N

• j=1

φ′′(λj) − βN

N

• j=1

V ′(λj)φ′(λj) + β

N

• i,j=1

i=j

φ′(λi) λi − λj L f (λ) =

• 1 − β

2

• N

ˆ φ′′dµV + βN ˆ Ξ(φ′)dνN +

• 1 − β

2 ˆ φ′′dνN + β 2 ¨ φ′(x) − φ′(y) x − y νN(dx)νN(dy)

• = R(λ)

.

β-ensembles

The approximate eigenfunction property

So if we suppose that φ satisfies the equation Ξ(φ′) = −κφ, κ > 0, we obtain L ˆ φ dνN

• = −Λ

ˆ φ dνN −

• 1

2 − 1 β

• m(φ)
• + R(λ),

with Λ = Nβκ. That is, the random variable ˆ φdνN −

• 1

2 − 1 β

• m(φ) is an

approximate eigenfunction of the generator L .

β-ensembles

The approximate eigenfunction property

So if we suppose that φ satisfies the equation Ξ(φ′) = −κφ, κ > 0, we obtain L ˆ φ dνN

• = −Λ

ˆ φ dνN −

• 1

2 − 1 β

• m(φ)
• + R(λ),

with Λ = Nβκ. That is, the random variable ˆ φdνN −

• 1

2 − 1 β

• m(φ) is an

approximate eigenfunction of the generator L . Moreover, |∇f (λ)|2 = N

j=1 φ′(λj)2 and if we let

σ2 = lim

N→∞

1 ΛEN

V ,β

• |∇f |2

= 1 βκ ˆ |φ′|2dµV we obtain S(λ) =

N

• j=1

φ′(λj)2 − Λσ2 = ˆ |φ′|2dνN. Hence, if β = 2, we have the bound: W1 ˆ φdνN, σZ

• ≤ EN

V ,β|R| + EN V ,β|S|

Nβκ .

β-ensembles

Normal approximation in the GUE case

If f is any polynomial, then EN

GUE

• ˆ

fdνN

• 2

≤ Cf .

β-ensembles

Normal approximation in the GUE case

If f is any polynomial, then EN

GUE

• ˆ

fdνN

• 2

≤ Cf . This implies that if φ is a polynomial, EN

GUE

• |R(λ)|
• ≤ EN

GUE

• ˆ

φ′′dνN

• + EN

GUE

• ¨ φ′(x) − φ′(y)

x − y νN(dx)νN(dy)

• = O(1)

N→∞

EN

GUE

• |S(λ)|
• = EN

GUE

• ˆ

|φ′|2dνN

• = O(1)

N→∞

.

β-ensembles

Normal approximation in the GUE case

If f is any polynomial, then EN

GUE

• ˆ

fdνN

• 2

≤ Cf . This implies that if φ is a polynomial, EN

GUE

• |R(λ)|
• ≤ EN

GUE

• ˆ

φ′′dνN

• + EN

GUE

• ¨ φ′(x) − φ′(y)

x − y νN(dx)νN(dy)

• = O(1)

N→∞

EN

GUE

• |S(λ)|
• = EN

GUE

• ˆ

|φ′|2dνN

• = O(1)

N→∞

. Therefore, if the polynomial φ satisfies Ξ(φ′) = −κφ , we conclude that W1 ˆ φdνN, σZ

• =

O

N→∞(N−1)

where σ2 = 1 βκ ˆ φ′(x)2µsc(dx) = − 1 β ˆ φ′(x)Ξ−1(φ)(x)µsc(dx) = 2 β Σ(φ).

β-ensembles

Chebyshev polynomials

If the equilibrium is the semicircle law, Ξx(ψ) = −2xψ(x) + 2 π ˆ 1

−1

ψ(x) − ψ(t) x − t

• 1 − x2dx.

For any ψ ∈ L2(µsc), we have for all x ∈ [−1, 1], Ξx(ψ) = ˆ K(x, t)ψ(t)dt, K(x, t) = −2

∞

• k=1

Tk(x)Uk−1(t), where (Uk)∞

k=0 are the Chebyshev polynomials of the second kind:

Uk−1 = k−1T ′

k.

Since (Uk)∞

k=0 is an orthogonal basis of L2(µsc), this shows that for all k ≥ 1,

Ξx(T ′

k) = −2kTk(x),

x ∈ [−1, 1].

β-ensembles

Conclusion

The asymptotic variance of the statistic ˆ TkdνN is σ2

k = 1

4k ˆ T ′

k(x)2µsc(dx)

= k 4 ˆ Uk−1(x)2µsc(dx) = k 4 .

β-ensembles

Conclusion

The asymptotic variance of the statistic ˆ TkdνN is σ2

k = 1

4k ˆ T ′

k(x)2µsc(dx)

= k 4 ˆ Uk−1(x)2µsc(dx) = k 4 . Let (Zn)n≥1 be i.i.d. N(0, 1). For any d ∈ N, we obtain W1 ˆ TndνN d

n=1,

√n 2 Zn d

n=1

• = Od

N→∞

• N−1

.

β-ensembles

Conclusion

The asymptotic variance of the statistic ˆ TkdνN is σ2

k = 1

4k ˆ T ′

k(x)2µsc(dx)

= k 4 ˆ Uk−1(x)2µsc(dx) = k 4 . Let (Zn)n≥1 be i.i.d. N(0, 1). For any d ∈ N, we obtain W1 ˆ TndνN d

n=1,

√n 2 Zn d

n=1

• = Od

N→∞

• N−1

. Corollary For any real–valued polynomial Q, W1 ˆ Q dνN, Σ(Q)Z

• ≤ CQN−1,

where Σ2(Q) =

∞

• n=1
• Q2

nσ2 n = 1

4

∞

• n=1

n Q2

n.

β-ensembles

Thank you!