Motivation
Conjecture
General approach
Result in a particular case
Conclusion
A Markovian approach of the Central Limit Theorem
Claire Delplancke
Institut de Mathématiques de Toulouse
July 13 2015
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
1 Motivation
2 Conjecture
3 General approach
4 Result in a particular case
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
1 Motivation
2 Conjecture
3 General approach
4 Result in a particular case
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Central limit theorem
X1 , . . . Xn iid real rv with E[X1 ] = 0, E[X12 ] = 1.
γ the standard gaussian measure.
n
1 X
Yn = √
Xk
n
→
γ.
k=1
(Yn )n≥1 is an inhomogenous Markov chain.
n−1
Yn
1
1 X
Xi + √ Xn
= √
n
n
i=1
r
n−1
1
=
Yn−1 + √ Xn .
n
n
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Goal
Hypothesis and notation
γ the standard gaussian measure
ϕ ∈ L2 (γ) density of X1 wrt γ
fn ∈ L2 (γ) density of Yn wrt γ.
Goal : Prove
Z
Varγ (fn ) =
Claire Delplancke
A Markovian approach of the Central Limit Theorem
(fn − 1)2 d γ → 0.
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Berry-Esseen-like theorems
Z
Z
dF (µ, ν) = sup fd µ − fd ν .
f ∈F
Name
Definition
Rate of convergence
Kolmogorov
Total variation
F = 1]−∞,x]
F = {|f |∞ ≤ 1}
Wasserstein
F = Lip1
Relative entropy
Entγ (ν) =
n−1/2 (Berry Esseen ’48)
n−1/2
(Sirazhdinov Mamatov ’62,
Bally Caramellino ’14)
n−1/2
(Ibragimov ’66, Rio ’11)
n−1
(Arstein et al. ’04)
?
1/2
Varγ
R
dν
dγ
dν
log dγ
dγ
F = unit ball of L2 (γ)
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
More on Varµ
Z
Z
d ν 1/2
Varµ ( ) = sup fd ν − f µ
dµ
kf kL2 (µ) ≤1
actually not a distance between µ and ν as F = unit ball of L2 (µ)
depends of µ.
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Comparison between distances
Pinsker dTV (ν, µ) ≤ (2Entµ (ν))1/2
dKol ≤ dTV
1/2
dKol ≤ CdW under additional hyp.
dν 1/2
By inclusion of classes F dTV (ν, µ) ≤ Varµ ( dµ
)
dν 1/2
By inclusion of classes F dW (ν, µ) ≤ Varµ ( dµ
)
dν
As ∀x ≥ 0 log x ≤ (x − 1) then Entµ (ν) ≤ Varµ ( dµ
)
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Motivation
Markovian structure
→ linear operators on L2 (γ)
→ nicely combines with L2 -functional
f ∈ L2 (γ) : not much regularity assumptions.
But strong assumption on moments : ϕ ∈ L2 (γ) implies moment of
every order exists.
Similar approach used succesfully in [Arnaudon Miclo 2013] for
finding Fréchet means on Riemannian manifolds.
CLT = example of an inhomogenous Markov process → stochastic
algorithms.
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
1 Motivation
2 Conjecture
3 General approach
4 Result in a particular case
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Conjecture I
Conjecture
Let (Xn )n≥1 be iid real rv.
Suppose X1 has a density ϕ wrt γ and ϕ ∈ L2 (γ).
If E[X1 ] = 0, E[X12 ] = 1 then ∃c > 0 such that
Varγ (fn ) ≤
Claire Delplancke
A Markovian approach of the Central Limit Theorem
c
.
n
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Conjecture II
Generalized conjecture
Let (Xn )n≥1 be iid real rv.
Suppose X1 has a density ϕ wrt γ andRϕ ∈ L2 (γ).
For r ≥ 2 natural number, if E[X1k ] = x k d γ for k = 0, . . . , r , then
∃c > 0 such that
c
Varγ (fn ) ≤ r −1 .
n
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
1 Motivation
2 Conjecture
3 General approach
4 Result in a particular case
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
The Markov operator and the dual operator
The family of operators
Qp,q g (y ) := E [g (Yq )|Yp = y ]
(q ≥ p)
is an inhomogenous Markov semi-group :
Qp,q (1) = 1.
Qp,r = Qp,q Qq,r
(r ≥ q ≥ p).
∗
∗
fn−1 .
If Qn−1,n
is the dual in L2 (γ) of Qn−1,n , then fn = Qn−1,n
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Proof
Indeed
Z
E[g (Yn )]
E[g (Yn )]
=
Markov property
=
=
=
Claire Delplancke
A Markovian approach of the Central Limit Theorem
gfn d γ
E[(Qn−1,n g )(Yn−1 )]
Z
(Qn−1,n g )fn−1 d γ
Z
∗
fn−1 )d γ.
g (Qn−1,n
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Expression of the Markov operator
Recall
γ standard gaussian measure.
ϕ ∈ L2 (γ) density of X1 w.r.t. γ.
Yn =
q
1 − n1 Yn−1 +
√1 Xn .
n
For every ”good” function f
r
Z
Qn−1,n f (y ) =
f
Claire Delplancke
A Markovian approach of the Central Limit Theorem
1
1
1− y + √ x
n
n
!
ϕ(x)d γ(x).
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Expression of the Markov operator and the dual operator
Simplify notation :
Z
Qn−1,n f (y ) =
q
1−
1
n
= e −tn and
√1
n
=
√
1 − e −2tn = e −t̃n .
p
f e −tn y + 1 − e −2tn x ϕ(x)d γ(x)
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Expression of the Markov operator and the dual operator
Simplify notation :
(x) =
1
n
= e −tn and
√1
n
=
√
1 − e −2tn = e −t̃n .
Z
p
f e −tn y + 1 − e −2tn x ϕ(x)d γ(x)
Z
f (e −tn x +
Qn−1,n f (y ) =
∗
Qn−1,n
f
q
1−
p
p
1 − e −2tn y )ϕ( 1 − e −2tn x − e −tn y )d γ(y )
∗
∗
Remark Qn−1,n
is non-conservative : Qn−1,n
1 6= 1.
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Link between the Ornstein-Uhlenbeck semi-group and the
dual operator I
Observe the similarity with the Ornstein-Uhlenbeck operator Pt
Z
p
p
∗
Qn−1,n
f (x) =
f (e −tn x + 1 − e −2tn y )ϕ( 1 − e −2tn x − e −tn y )d γ(y )
Z
p
Pt f (x) =
f (e −t x + 1 − e −2t y )d γ(y ).
∗
f = Ptn f .
If ϕ = 1 then Qn−1,n
∗
If f = 1 then Qn−1,n
1 = Pt̃n ϕ.
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Link between the Ornstein-Uhlenbeck semi-group and the
dual operator II
Remark 1 We can show that for every function f sufficiently regular (e.g.
f ∈ C3b )
Qn−1,n f
∼ Ptn f + O(n−3/2 )
∗
Qn−1,n
f
∼ Ptn f + O(n−3/2 ).
Remark 2 : no sufficient regularity assuption on f to control the
remainder term.
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Outline of the approach I
For any density function f ∈ L2 (γ) we need to upper-bound
Z
∗
∗
Varγ (Qn−1,n f ) = (Qn−1,n
f − 1)2 d γ
in terms of
Z
Varγ (f ) =
Claire Delplancke
A Markovian approach of the Central Limit Theorem
(f − 1)2 d γ.
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Outline of the approach I
For any density function f ∈ L2 (γ) we need to upper-bound
Z
∗
∗
Varγ (Qn−1,n f ) = (Qn−1,n
f − 1)2 d γ
in terms of
Z
Varγ (f ) =
(f − 1)2 d γ.
∗
∗
Remark Qn−1,n
f − 1 6= Qn−1,n
(f − 1).
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Outline of the approach II
Set g = f − 1.
∗
Varγ (Qn−1,n
f)
Z
=
≤
Cauchy
-Schwarz
∗
∗
(Qn−1,n
g + Qn−1,n
1 − 1)2 d γ
Z
Z
∗
∗
2
g )2 d γ
(Qn−1,n 1 − 1) d γ + (Qn−1,n
{z
} |
{z
}
|
(1)
(2)
p
+2 (1) (2).
Outline :
Estimate (1)
Estimate (2)
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Tool : Hermite polynomials
1
2
2
H̄k (x) = (−1)k √ e x /2 ∂xk e −x /2 .
k!
Degree of H̄k is k, H̄0 = 1.
Orthonormal basis of L2 (γ) : if g ∈ L2 (γ) then
Z
X
X
g=
gk H̄k
g 2d γ =
gk2 .
k≥0
k≥0
Diagonalize Pt the Ornstein-Uhlenbeck
semi-group
Pt (H̄k ) = e −kt H̄k .
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Upper bound of (1)
Decomposition of Pt on Hermite polynomials ⇒
Z
∗
(1) =
(Qn−1,n
1 − 1)2 d γ
Z
=
(Pt̃n ϕ − 1)2 d γ
Z
−6t̃n
≤ e
(ϕ − 1)2 d γ.
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Upper bound of (2) I
Z
∗
(Qn−1,n
g )2 d γ
Z
?
≤ Cn g 2 d γ.
(2) =
∗
in the (H̄k ) basis : matrix A = Aϕ
Write the (infinite) matrix of Qn−1,n
n
depending on
Time n
Coefficients ϕk of ϕ along the (H̄k ).
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Upper bound of (2) II
Z
∗
(Qn−1,n
g )2 d γ =
X
∗
Qn−1,n
g
2
k
k≥0
=
X
(Ag )2k
k≥0
= hAg , Ag iL2 (RN )
= hg , AT Ag iL2 (RN ) .
sup |λ|, λ eigenvalue of AT A
=?
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
1 Motivation
2 Conjecture
3 General approach
4 Result in a particular case
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Choose ϕ =
Conjecture
P
General approach
Result in a particular case
Conclusion
n≥0 ϕn H̄n .


ϕd
γ
=
1

 ϕ0 = 1
R
γ=0
ϕ1 = 0
⇔
R xϕd


x 2 ϕd γ = 1
ϕ2 = 0
R
ϕ ≥ 0 → choose ϕ = H̄0 + ϕ4 H̄4 .
Lemma
ϕ = H̄0 + ϕ4 H̄4 is a density iff
√
0 ≤ ϕ4 ≤
Then E[X1k ] =
R
6
.
3
x k d γ for k = 0, . . . , 3.
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Particular case
Choose ϕ = H̄0 + C H̄4 .
AT A non zero only on coeff (i, i), (i, i − 4), (i, i + 4)
√
Under hypothesis ϕ4 ≤ 36
we derive an upper-bound for sup |λ|, λ eigenvalue of AT A
We complete the estimation of (2)
(2) ≤ (1 − 8tn +
Claire Delplancke
A Markovian approach of the Central Limit Theorem
O(tn2 ))
Z
g 2 d γ.
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Particular case
Putting all pieces back together, the conjecture holds in the particular
case :
Theorem
If ϕ = H̄0 + C H̄4 , there exists a constant c such that for all big enough n,
Varγ (fn ) ≤ Varγ (ϕ)
Claire Delplancke
A Markovian approach of the Central Limit Theorem
c
.
n2
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Difficulties of general case
Matrix A depends on
coefficients (ϕk )
time n
Knowledge over (ϕk )
ϕ0 = 1, ϕ1 = · · · = ϕr = 0
P 2
ϕk < ∞
(ϕk )k≥0 coefficients of a positive function.
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Conclusion
Proof to be completed
In a particular case satisfying E[X13 ] = 0
Varγ (fn ) ≤ Varγ (ϕ)
c
.
n2
dTV (L(Yn ), γ) ≤ Varγ (fn )1/2 and if E[X13 ] = 0 rate
optimal.
1
n
for dTV
Extension to other stochastic algorithms.
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse
Motivation
Conjecture
General approach
Result in a particular case
Conclusion
Thank you !
Claire Delplancke
A Markovian approach of the Central Limit Theorem
Institut de Mathématiques de Toulouse