Introduction
Least squares estimation
Main results
On the asymptotic behavior of bifurcating
autoregressive processes
B. Bercu, B. De Saporta, and A. Gégout-Petit
Institut de Mathématiques de Bordeaux, France
Séminaire de Statistique de l’IRMAR
Université de Rennes 1, 11 décembre 2009
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
1 / 28
Introduction
Least squares estimation
Main results
Outline
1
Introduction
Bifurcating autoregressive processes
The state of the art
2
Least squares estimation
Martingale assumptions
Our least squares estimators
3
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
2 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Outline
1
Introduction
Bifurcating autoregressive processes
The state of the art
2
Least squares estimation
Martingale assumptions
Our least squares estimators
3
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
3 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Cell division
1
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
4 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Cell division
2
1
3
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
4 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Cell division
4
2
5
1
6
3
7
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
4 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Cell division
8
4
9
2
10
5
11
1
12
6
13
3
14
7
15
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
4 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Ancestors and offsprings
The offsprings of cell n are labelled
2n,
2n + 1.
2n
For each cell n, Xn can be
n
2n+1
the diameter,
the age,...
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
5 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Bifurcating AutoRegressive processes
Consider the BAR(p) process given, for all n > 2p−1 , by

p
X



bk X[ kn−1 ] + ε2n ,
= a +

 X2n
2
k =1
p
X




X
=
c
+
dk X[ kn−1 ] + ε2n+1
 2n+1
2
k =1
where [x] stands for the integer part of x and (ε2n , ε2n+1 ) is the
driven noise of the process.
Goal
Statistical inference on the unknown parameters (a, b, c, d).
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
6 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Bifurcating AutoRegressive processes
For simplicity, we focus on the BAR(1) process given, for all
n > 1, by

 X2n
= a + b Xn + ε2n ,
 X
2n+1 = c + d Xn + ε2n+1
where 0 < max(|b|, |d|) < 1 and |a| + |c| =
6 0.
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
7 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Binary tree, generations, subtrees
Generation 0 is G0 = {1}.
8
Generation 1 is G1 = {2, 3}.
9
Generation 2 is G2 = {4, 5, 6, 7}.
4
2
10
Generation n is
11
Gn = {2n , 2n + 1, . . . , 2n+1 − 1}.
12
Subtree up to time n is
5
1
6
13
Tn =
3
14
n
[
Gk .
k =0
Cardinalities are |Gn | = 2n and
7
15
|Tn | = 2n+1 − 1.
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
logobordeaux
8 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Binary tree, generations, subtrees
Generation 0 is G0 = {1}.
8
Generation 1 is G1 = {2, 3}.
9
Generation 2 is G2 = {4, 5, 6, 7}.
4
2
10
Generation n is
11
Gn = {2n , 2n + 1, . . . , 2n+1 − 1}.
12
Subtree up to time n is
5
1
6
13
Tn =
3
14
n
[
Gk .
k =0
Cardinalities are |Gn | = 2n and
7
15
|Tn | = 2n+1 − 1.
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
logobordeaux
8 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Binary tree, generations, subtrees
Generation 0 is G0 = {1}.
8
Generation 1 is G1 = {2, 3}.
9
Generation 2 is G2 = {4, 5, 6, 7}.
4
2
10
Generation n is
11
Gn = {2n , 2n + 1, . . . , 2n+1 − 1}.
12
Subtree up to time n is
5
1
6
13
Tn =
3
14
n
[
Gk .
k =0
Cardinalities are |Gn | = 2n and
7
15
|Tn | = 2n+1 − 1.
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
logobordeaux
8 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Binary tree, generations, subtrees
Generation 0 is G0 = {1}.
8
Generation 1 is G1 = {2, 3}.
9
Generation 2 is G2 = {4, 5, 6, 7}.
4
2
10
Generation n is
11
Gn = {2n , 2n + 1, . . . , 2n+1 − 1}.
12
Subtree up to time n is
5
1
6
13
Tn =
3
14
n
[
Gk .
k =0
Cardinalities are |Gn | = 2n and
7
15
|Tn | = 2n+1 − 1.
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
logobordeaux
8 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Binary tree, generations, subtrees
Generation 0 is G0 = {1}.
2n
Generation 1 is G1 = {2, 3}.
Generation 2 is G2 = {4, 5, 6, 7}.
Generation n is
Gn = {2n , 2n + 1, . . . , 2n+1 − 1}.
Subtree up to time n is
Tn =
n
[
Gk .
k =0
Cardinalities are |Gn | = 2n and
|Tn | = 2n+1 − 1.
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
logobordeaux
8 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Binary tree, generations, subtrees
Generation 0 is G0 = {1}.
2n
Generation 1 is G1 = {2, 3}.
Generation 2 is G2 = {4, 5, 6, 7}.
Generation n is
Gn = {2n , 2n + 1, . . . , 2n+1 − 1}.
Subtree up to time n is
Tn =
n
[
Gk .
k =0
Cardinalities are |Gn | = 2n and
|Tn | = 2n+1 − 1.
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
logobordeaux
8 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Filtration
8
4
9
2
10
5
At time 0, F0 = σ{X1 }.
At time 1, F1 = σ{X1 , X2 , X3 }.
At time 2,
11
1
12
F2 = σ{X1 , X2 , X3 , X4 , X5 , X6 , X7 }.
13
At time n,
6
3
14
Fn = σ{Xk with k ∈ Tn }.
7
15
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
9 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Filtration
8
4
9
2
10
5
11
1
12
At time 0, F0 = σ{X1 }.
At time 1, F1 = σ{X1 , X2 , X3 }.
At time 2,
F2 = σ{X1 , X2 , X3 , X4 , X5 , X6 , X7 }.
6
13
At time n,
3
14
Fn = σ{Xk with k ∈ Tn }.
7
15
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
9 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Filtration
8
4
9
2
10
5
11
1
12
At time 0, F0 = σ{X1 }.
At time 1, F1 = σ{X1 , X2 , X3 }.
At time 2,
F2 = σ{X1 , X2 , X3 , X4 , X5 , X6 , X7 }.
6
13
At time n,
3
14
Fn = σ{Xk with k ∈ Tn }.
7
15
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
9 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Filtration
2n
At time 0, F0 = σ{X1 }.
At time 1, F1 = σ{X1 , X2 , X3 }.
At time 2,
F2 = σ{X1 , X2 , X3 , X4 , X5 , X6 , X7 }.
At time n,
Fn = σ{Xk with k ∈ Tn }.
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
9 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Previous works I
Stationary processes
Cowan,Staudte, Biometrics, 1986 : Introduction, motivation
in Statistical Biology.
Huggins, Annals of Statistic, 1996 : Maximum likelihood
estimation for large trees.
Huggins, Basawa, Journal of Applied Probability, 1999 and
Australian Journal of Statistics, 2000 : Maximum likelihood
estimation for large trees and higher order.
Basawa, Zhou, Journal of Applied Probability, 2004 :
exponential BAR processes, and Journal of Time Series
Analysis, 2005 : CLT for BAR processes.
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
10 / 28
Introduction
Least squares estimation
Main results
Bifurcating autoregressive processes
The state of the art
Previous works II
Non stationary processes
Guyon, Annals of Applied Probability, 2007 : Least square
estimation for Gaussian BAR processes via a Markov
chain approach.
Delmas, Marsalle, 2008, 2009 : The same as Guyon and
possibility for a cell to die.
Our work, EJP 2009 : Least square estimation for
martingale difference BAR(p) processes by use of a
martingale approach.
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
11 / 28
Introduction
Least squares estimation
Main results
Martingale assumptions
Our least squares estimators
Outline
1
Introduction
Bifurcating autoregressive processes
The state of the art
2
Least squares estimation
Martingale assumptions
Our least squares estimators
3
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
12 / 28
Introduction
Least squares estimation
Main results
Martingale assumptions
Our least squares estimators
Martingale assumptions
(H.1) ∀n > 0 and ∀k ∈ Gn+1 ,
E[εk |Fn ] = 0
and
E[ε2k |Fn ] = σ 2 > 0
a.s.
(H.2) ∀n > 0 and ∀k 6= ` ∈ Gn+1 ,
• if [k /2] 6= [`/2], then εk and ε` are independent given Fn
• if [k /2] = [`/2], then for ρ < σ 2
E[εk ε` |Fn ] = ρ
a.s.
(H.3)
sup sup E[ε4k |Fn ] < ∞
a.s.
n>0 k ∈Gn+1
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
13 / 28
Introduction
Least squares estimation
Main results
Martingale assumptions
Our least squares estimators
Martingale assumptions
(H.1) ∀n > 0 and ∀k ∈ Gn+1 ,
E[εk |Fn ] = 0
and
E[ε2k |Fn ] = σ 2 > 0
a.s.
(H.2) ∀n > 0 and ∀k 6= ` ∈ Gn+1 ,
• if [k /2] 6= [`/2], then εk and ε` are independent given Fn
• if [k /2] = [`/2], then for ρ < σ 2
E[εk ε` |Fn ] = ρ
a.s.
(H.3)
sup sup E[ε4k |Fn ] < ∞
a.s.
n>0 k ∈Gn+1
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
13 / 28
Introduction
Least squares estimation
Main results
Martingale assumptions
Our least squares estimators
Martingale assumptions
(H.1) ∀n > 0 and ∀k ∈ Gn+1 ,
E[εk |Fn ] = 0
and
E[ε2k |Fn ] = σ 2 > 0
a.s.
(H.2) ∀n > 0 and ∀k 6= ` ∈ Gn+1 ,
• if [k /2] 6= [`/2], then εk and ε` are independent given Fn
• if [k /2] = [`/2], then for ρ < σ 2
E[εk ε` |Fn ] = ρ
a.s.
(H.3)
sup sup E[ε4k |Fn ] < ∞
a.s.
n>0 k ∈Gn+1
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
13 / 28
Introduction
Least squares estimation
Main results
Martingale assumptions
Our least squares estimators
Estimation of the parameter
We estimate θ = (a, b, c, d)t by the
Least squares estimator


bn
a
X2k
X  X X
bn 
 b
k 2k
 = (I2 ⊗ S −1 )

= 
n−1
 X2k +1
 b
cn 
k ∈Tn−1
bn
Xk X2k +1
d

θbn




where
Sn =
X 1
Xk
k ∈Tn
Bercu, De Saporta, Gégout-Petit
Xk
Xk2
.
On bifurcating autoregressive processes
logobordeaux
14 / 28
Introduction
Least squares estimation
Main results
Martingale assumptions
Our least squares estimators
A martingale identity

ε2k
X  X ε

k 2k 
−1

(θbn − θ) = (I2 ⊗ Sn−1
)
 ε2k +1  .
k ∈Tn−1
Xk ε2k +1

Consequently, if Σn = I2 ⊗ Sn , we have the martingale identity
θbn − θ = Σ−1
n−1 Mn
where Mn is the martingale given by

ε2k
X  X ε
k 2k

Mn =
 ε2k +1
k ∈Tn−1
Xk ε2k +1
Bercu, De Saporta, Gégout-Petit


.

On bifurcating autoregressive processes
logobordeaux
15 / 28
Introduction
Least squares estimation
Main results
Martingale assumptions
Our least squares estimators
Estimation of the conditional variance and covariance
We estimate the conditional variance σ 2 and covariance ρ by
Conditional variance estimator
σ
bn2 =
X
1
(b
ε22k + εb22k +1 )
2|Tn−1 |
k ∈Tn−1
Conditional covariance estimator
X
1
ρbn =
εb2k εb2k +1
|Tn−1 |
k ∈Tn−1
where for all k ∈ Gn
(
bn Xk ,
bn − b
εb2k = X2k − a
bn Xk .
εb2k +1 = X2k +1 − b
cn − d
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
logobordeaux
16 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
Outline
1
Introduction
Bifurcating autoregressive processes
The state of the art
2
Least squares estimation
Martingale assumptions
Our least squares estimators
3
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
17 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
Strong laws for the noise
If (εn ) satisfies (H.1), (H.2) and (H.3), we have
1 X
εk
n→+∞ |Tn |
lim
= 0
a.s.
k ∈Tn
1 X 2
lim
εk = σ 2
n→+∞ |Tn |
a.s.
k ∈Tn
lim
n→+∞
1
|Tn−1 |
X
ε2k ε2k +1 = ρ
a.s.
k ∈Tn−1
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
18 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
Some notations
First of all, let
a=
a+c
,
2
b=
b+d
,
2
ab + cd
a2 + c 2
b2 + d 2
,
a2 =
,
b2 =
.
2
2
2
In addition, denote by Γ and L the two symmetric matrices
!
!
σ2 ρ
1 λ
Γ=
and
L=
λ `
ρ σ2
ab =
where
λ=
a
1−b
Bercu, De Saporta, Gégout-Petit
and
`=
a2 + σ 2 + 2λab
1 − b2
On bifurcating autoregressive processes
.
logobordeaux
19 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
A keystone lemma
Lemma
If (εn ) satisfies (H.1), (H.2) and (H.3), we have
lim
Sn
n→∞
|Tn |
=L
a.s.
Limiting matrix
The limiting matrix L is positive definite.
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
20 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
Strong law for the parameter
Theorem
Assume that (εn ) satisfies (H.1), (H.2) and (H.3). Then, θbn
converges a.s. to θ with the rate of convergence
log |Tn−1 |
2
b
k θn − θ k = O
a.s.
|Tn−1 |
In addition, if Λ = I2 ⊗ L, we have the quadratic strong law
lim
n→∞
n
1X
n
|Tk −1 |(θbk − θ)t Λ(θbk − θ) = 4σ 2
a.s.
k =1
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
21 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
Tools for the strong laws
The strong laws are related to the random variable
Vn = (θbn − θ)t Σn−1 (θbn − θ)
where Σn = I2 ⊗ Sn .
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
22 / 28
A keystone lemma
Strong laws of large numbers
Central limit theorems
Introduction
Least squares estimation
Main results
Tools for the strong laws
We have the main decomposition
Vn+1 + An = V1 + Bn+1 + Wn+1 ,
where, if ∆Mn+1 = Mn+1 − Mn ,
An =
n
X
−1
Mkt (Σ−1
k −1 − Σk )Mk ,
k =2
Bn+1 = 2
n
X
Mkt Σ−1
k ∆Mk +1 ,
k =2
Wn+1 =
n
X
∆Mkt +1 Σ−1
k ∆Mk +1 .
logobordeaux
k =2
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
23 / 28
A keystone lemma
Strong laws of large numbers
Central limit theorems
Introduction
Least squares estimation
Main results
Tools for the strong laws
We have the main decomposition
Vn+1 + An = V1 + Bn+1 + Wn+1 ,
where, if ∆Mn+1 = Mn+1 − Mn ,
An =
n
X
−1
Mkt (Σ−1
k −1 − Σk )Mk ,
k =2
Bn+1 = 2
n
X
Mkt Σ−1
k ∆Mk +1 ,
k =2
Wn+1 =
n
X
∆Mkt +1 Σ−1
k ∆Mk +1 .
logobordeaux
k =2
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
23 / 28
A keystone lemma
Strong laws of large numbers
Central limit theorems
Introduction
Least squares estimation
Main results
Tools for the strong laws
We have the main decomposition
Vn+1 + An = V1 + Bn+1 + Wn+1 ,
where, if ∆Mn+1 = Mn+1 − Mn ,
An =
n
X
−1
Mkt (Σ−1
k −1 − Σk )Mk ,
k =2
Bn+1 = 2
n
X
Mkt Σ−1
k ∆Mk +1 ,
k =2
Wn+1 =
n
X
∆Mkt +1 Σ−1
k ∆Mk +1 .
logobordeaux
k =2
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
23 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
Strong law for the conditional variance
Denote
σn2 =
X
1
(ε22k + ε22k +1 ).
2|Tn−1 |
k ∈Tn−1
Theorem
b n2
Assume that (εn ) satisfies (H.1), (H.2) and (H.3). Then, σ
2
converges a.s. to σ ,
1 X
(b
ε2k − ε2k )2 + (b
ε2k +1 − ε2k +1 )2 = 4σ 2
n→∞ n
lim
a.s.
k ∈Tn−1
lim
n→∞
Bercu, De Saporta, Gégout-Petit
|Tn |
n
(b
σn2 − σn2 ) = 4σ 2
a.s.
logobordeaux
On bifurcating autoregressive processes
24 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
Strong law for the conditional covariance
Denote
ρn =
1
|Tn−1 |
X
ε2k ε2k +1 .
k ∈Tn−1
Theorem
Assume that (εn ) satisfies (H.1), (H.2) and (H.3). Then, ρbn
converges a.s. to ρ,
1 X
(b
ε2k − ε2k )(b
ε2k +1 − ε2k +1 ) = 2ρ
n→∞ n
lim
a.s.
k ∈Tn−1
lim
n→∞
Bercu, De Saporta, Gégout-Petit
|Tn |
n
(ρbn − ρn ) = 4ρ
a.s.
logobordeaux
On bifurcating autoregressive processes
25 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
More assumptions
(H.4) ∀n > 0 and ∀k ∈ Gn+1 ,
E[ε4k |Fn ] = τ 4
a.s.
Moreover, ∀k 6= ` ∈ Gn+1 with [k /2] = [`/2] and for ν 2 < τ 4
E[ε2k ε2` |Fn ] = ν 2
a.s.
(H.5)
sup sup E[ε8k |Fn ] < ∞
a.s.
n>0 k ∈Gn+1
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
26 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
More assumptions
(H.4) ∀n > 0 and ∀k ∈ Gn+1 ,
E[ε4k |Fn ] = τ 4
a.s.
Moreover, ∀k 6= ` ∈ Gn+1 with [k /2] = [`/2] and for ν 2 < τ 4
E[ε2k ε2` |Fn ] = ν 2
a.s.
(H.5)
sup sup E[ε8k |Fn ] < ∞
a.s.
n>0 k ∈Gn+1
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
26 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
Central limit theorems
Theorem
If (εn ) satisfies (H.1) to (H.5), we have
q
L
|Tn−1 |(θbn − θ) −→ N (0, Γ ⊗ L−1 ),
q
τ 4 − 2σ 4 + ν 2 L
,
|Tn−1 |(b
σn2 − σ 2 ) −→ N 0,
2
q
L
|Tn−1 |(ρbn − ρ) −→ N (0, ν 2 − ρ2 ).
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
27 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
Central limit theorems
Theorem
If (εn ) satisfies (H.1) to (H.5), we have
q
L
|Tn−1 |(θbn − θ) −→ N (0, Γ ⊗ L−1 ),
q
τ 4 − 2σ 4 + ν 2 L
,
|Tn−1 |(b
σn2 − σ 2 ) −→ N 0,
2
q
L
|Tn−1 |(ρbn − ρ) −→ N (0, ν 2 − ρ2 ).
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
27 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
Central limit theorems
Theorem
If (εn ) satisfies (H.1) to (H.5), we have
q
L
|Tn−1 |(θbn − θ) −→ N (0, Γ ⊗ L−1 ),
q
τ 4 − 2σ 4 + ν 2 L
,
|Tn−1 |(b
σn2 − σ 2 ) −→ N 0,
2
q
L
|Tn−1 |(ρbn − ρ) −→ N (0, ν 2 − ρ2 ).
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
27 / 28
Introduction
Least squares estimation
Main results
A keystone lemma
Strong laws of large numbers
Central limit theorems
!!!! Many thanks for your attention !!!!
logobordeaux
Bercu, De Saporta, Gégout-Petit
On bifurcating autoregressive processes
28 / 28