Quantitative Central Limit Theorems and Cumulants in Wiener chaos
Quantitative Central Limit Theorems and Cumulants in Wiener chaos Aline Bonami, Universit´e d’Orl´eans Simons Symposium, April 4, 2016
The model for a quantitative CLT Let Xj a sequence of independent identically distributed random variables such that E(X1 ) = 0, E(X12 ) = 1 and E(|X1 |3 ) is finite. If Vn :=
X1 + X2 + · · · + Xn √ , n
then Vn converges in distribution to N ∼ N (0, 1). Theorem Berry-Esseen 1942. There exists some universal constant C such that, for all real z, |P(Vn < z) − P(N < z)| ≤
C E(|X1 |3 ) √ . n
Kolmogorov Distance dKol (X , Y ) := supz |P(X < z) − P(Y < z)|. Total variation distance dTV (X , Y ) := sup |P(X ∈ A) − P(Y ∈ A)|. Theorem Bobkov, Chistyakov, G¨ otze 2013. For X1 absolutely continuous and under additional assumptions dTV (Vn , N) ≤
C E(|X1 |3 ) √ . n
The setting of Breuer Major Theorem. Let (X (j))j∈Z a stationary sequence of Centered Gaussian random variables such that E(X (j)X (j + k)) = ρ(k). Interested in n−1
1 X d √ F (X (k)) → N (0, σ 2 ) n k=0
for some functionals F with the speed of convergence. Particular case of Hermite polynomials 2 q 2 t q t2 d Hq (t) = (−1) e e− 2 . q dt Theorem Breuer-Major 1983. Let ρ ∈ `q (Z). Then P P (i) Var √1n n−1 H (X (k)) −→ σq2 := q! k∈Z ρ(k)q , k=0 q Pn−1 d 2 √1 (ii) k=0 Hq (X (k)) → N (0, σq ). n
Wiener chaos and Fourth Moment Approach
L2 (X ) := L2 (Ω, A, P) =
M
Hq
where Hq is the q−th Wiener chaos. P In our example, Hq generated by the Hq ( finite aj X (j)). In particular Hq (X (k)) belongs to the Wiener chaos Hq . Particular case of quantitative CLT for Fn , when Fn belongs to Hq . Fourth Moment Theorem (Nualart Peccati 2005.)Let {Fn : n ≥ 1} a sequence of random variables in Hq such that E[Fn2 ] = 1 for all n ≥ 1. Then Fn converges in distribution to N ∼ N (0, 1) if and only if E[Fn4 ] → 3.
Fourth moment and Cumulants Moreover (Nourdin, Peccati 2009) dKol (Fn , N) ≤
q E[Fn4 ] − 3.
E[N 4 ] = 3. Use of Stein’s Method: Y ∼ N (0, 1) if and only if E(f 0 (Y )) = E(Yf (Y )), f smooth. Let F a real-valued random variable, φF (t) = E[e itF ] its characteristic function. The jth cumulant of F , denoted by κj (F ), is dj κj (F ) = (−i)j j log φF (t)|t=0 . dt κj (N) = 0 except for j = 2. When E(F ) = 0 and E(F 2 ) = 1, then κ3 (F ) = E(F 3 ), κ4 (F ) = E(F 4 ) − 3. E[F
m+1
m X m ] − κm+1 (F ) = κs (F )E[F m+1−s ]. s −1 s=1
Estimates of cumulants Theorem B., Bierm´ e, Nourdin, Peccati (2012 for small q) There exists universal constants cs (q) such that, for q ≥ 2 and s > 4, whenever F is in the chaos Hq and satisfies E(F ) = 0, E(F 2 ) = 1, then s
|κs (F )| ≤ cs (q) [κ4 (F )] 4 .
Reminiscent of the bounds for moments given by hypercontractivity: E(|F |s ) ≤ (s − 1)sq/2 . All moments (and cumulants) are bounded in terms of the second one, but all cumulants (except for the second one) are bounded in terms of the fourth cumulant. Known for q = 2, with cs (q) =
(s−1)! . 2×3s/4
Speeds of convergence. Recall: dKol (F , N) ≤
q E[F 4 ] − 3.
“smooth” distance dsmooth (X , Y ) :=
sup kh00 k∞ ≤1
|E[h(X )] − E[h(Y )]| .
Theorem (BBNP). Let Fn in the chaos Hq with E(Fn ) = 0, E(Fn2 ) = 1 and E (Fn4 ) → 3. Then dsmooth (Fn , N) ≈ max |E [Fn3 ]|, (E [Fn4 ] − 3 }. Nourdin Peccati 2015: same rate of convergence for dTV .
Tools for the proofs
I
Wiener-Itˆo stochastic integrals.
I
Malliavin calculus and integration by parts.
I
Stein’s method.
The model for a quantitative CLT Let Xj a sequence of independent identically distributed random variables such that E(X1 ) = 0, E(X12 ) = 1 and E(|X1 |3 ) is finite. If Vn :=
X1 + X2 + · · · + Xn √ , n
then Vn converges in distribution to N ∼ N (0, 1). Theorem Berry-Esseen 1942. There exists some universal constant C such that, for all real z, |P(Vn < z) − P(N < z)| ≤
C E(|X1 |3 ) √ . n
Kolmogorov Distance dKol (X , Y ) := supz |P(X < z) − P(Y < z)|. Total variation distance dTV (X , Y ) := sup |P(X ∈ A) − P(Y ∈ A)|. Theorem Bobkov, Chistyakov, G¨ otze 2013. For X1 absolutely continuous and under additional assumptions dTV (Vn , N) ≤
C E(|X1 |3 ) √ . n
The setting of Breuer Major Theorem. Let (X (j))j∈Z a stationary sequence of Centered Gaussian random variables such that E(X (j)X (j + k)) = ρ(k). Interested in n−1
1 X d √ F (X (k)) → N (0, σ 2 ) n k=0
for some functionals F with the speed of convergence. Particular case of Hermite polynomials 2 q 2 t q t2 d Hq (t) = (−1) e e− 2 . q dt Theorem Breuer-Major 1983. Let ρ ∈ `q (Z). Then P P (i) Var √1n n−1 H (X (k)) −→ σq2 := q! k∈Z ρ(k)q , k=0 q Pn−1 d 2 √1 (ii) k=0 Hq (X (k)) → N (0, σq ). n
Wiener chaos and Fourth Moment Approach
L2 (X ) := L2 (Ω, A, P) =
M
Hq
where Hq is the q−th Wiener chaos. P In our example, Hq generated by the Hq ( finite aj X (j)). In particular Hq (X (k)) belongs to the Wiener chaos Hq . Particular case of quantitative CLT for Fn , when Fn belongs to Hq . Fourth Moment Theorem (Nualart Peccati 2005.)Let {Fn : n ≥ 1} a sequence of random variables in Hq such that E[Fn2 ] = 1 for all n ≥ 1. Then Fn converges in distribution to N ∼ N (0, 1) if and only if E[Fn4 ] → 3.
Fourth moment and Cumulants Moreover (Nourdin, Peccati 2009) dKol (Fn , N) ≤
q E[Fn4 ] − 3.
E[N 4 ] = 3. Use of Stein’s Method: Y ∼ N (0, 1) if and only if E(f 0 (Y )) = E(Yf (Y )), f smooth. Let F a real-valued random variable, φF (t) = E[e itF ] its characteristic function. The jth cumulant of F , denoted by κj (F ), is dj κj (F ) = (−i)j j log φF (t)|t=0 . dt κj (N) = 0 except for j = 2. When E(F ) = 0 and E(F 2 ) = 1, then κ3 (F ) = E(F 3 ), κ4 (F ) = E(F 4 ) − 3. E[F
m+1
m X m ] − κm+1 (F ) = κs (F )E[F m+1−s ]. s −1 s=1
Estimates of cumulants Theorem B., Bierm´ e, Nourdin, Peccati (2012 for small q) There exists universal constants cs (q) such that, for q ≥ 2 and s > 4, whenever F is in the chaos Hq and satisfies E(F ) = 0, E(F 2 ) = 1, then s
|κs (F )| ≤ cs (q) [κ4 (F )] 4 .
Reminiscent of the bounds for moments given by hypercontractivity: E(|F |s ) ≤ (s − 1)sq/2 . All moments (and cumulants) are bounded in terms of the second one, but all cumulants (except for the second one) are bounded in terms of the fourth cumulant. Known for q = 2, with cs (q) =
(s−1)! . 2×3s/4
Speeds of convergence. Recall: dKol (F , N) ≤
q E[F 4 ] − 3.
“smooth” distance dsmooth (X , Y ) :=
sup kh00 k∞ ≤1
|E[h(X )] − E[h(Y )]| .
Theorem (BBNP). Let Fn in the chaos Hq with E(Fn ) = 0, E(Fn2 ) = 1 and E (Fn4 ) → 3. Then dsmooth (Fn , N) ≈ max |E [Fn3 ]|, (E [Fn4 ] − 3 }. Nourdin Peccati 2015: same rate of convergence for dTV .
Tools for the proofs
I
Wiener-Itˆo stochastic integrals.
I
Malliavin calculus and integration by parts.
I
Stein’s method.
