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Introduction Goals Framework for the results Results Application Deviation inequalities and moderate deviation principle for Bifurcating Markov Chains. S. Valère BITSEKI PENDA joint work with (H. Djellout and A. Guillin) Université Blaise Pascal Laboratoire de Mathématiques Colloque Jeunes Probabilistes Statisticiens, CIRM 2012
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Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

May 24, 2018

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Page 1: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Deviation inequalities and moderatedeviation principle for Bifurcating Markov

Chains.

S. Valère BITSEKI PENDAjoint work with

(H. Djellout and A. Guillin)

Université Blaise PascalLaboratoire de Mathématiques

Colloque Jeunes Probabilistes Statisticiens, CIRM 2012

Page 2: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Outline

1 IntroductionMotivationThe model of bifurcating Markov chain

2 Goals

3 Framework for the results

4 Results

5 Application

Page 3: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Motivation

Guyon, J. Limit theorems for bifurcating markov chains.Application to the detection of cellular aging. Ann. Appl.Probab., (2007), Vol. 17, No. 5-6, pp. 1538-1569.

Escherichia Coli (E.Coli)

Page 4: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Figure: Cell division from E. J. Stewart and al.

Page 5: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Motivation

Guyon, J. Bize, A. Paul, G. Stewart, E.J. Delmas, J.F.Taddéi, F. Statistical study of cellular aging. CEMRACS2004 Proceedings, ESAIM Proceedings, (2005), 14, pp.100-114.

Page 6: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Motivation

First order bifurcating autoregressive process BAR(1)

Page 7: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Motivation

First order bifurcating autoregressive process BAR(1)

X

n

Page 8: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Motivation

First order bifurcating autoregressive process BAR(1)

L(X1) = ν, and ∀n ≥ 1,

X2n = α0Xn + β0 + ε2n

X2n+1 = α1Xn + β1 + ε2n+1,(1)

Page 9: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Motivation

First order bifurcating autoregressive process BAR(1)

where ν is a distribution probability on R, α0, α1 ∈ (−1, 1);β0, β1 ∈ R and

((ε2n, ε2n+1), n ≥ 1

)forms a sequence of

centered i.i.d bivariate random variables with covariancematrix

Γ = σ2(

1 ρρ 1

), σ2 > 0, ρ ∈ (−1, 1).

Page 10: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Motivation

θ = (α0, β0, α1, β1), σ and ρ.

H0 = (α0, β0) = (α1, β1) against H1 = (α0, β0) 6= (α1, β1).

Page 11: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Motivation

θ = (α0, β0, α1, β1), σ and ρ.H0 = (α0, β0) = (α1, β1) against H1 = (α0, β0) 6= (α1, β1).

Page 12: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Motivation

L(X1) = ν, and ∀n ≥ 1,

X2n = α0Xn + β0 + ε2n

X2n+1 = α1Xn + β1 + ε2n+1,(1)

where ν is a distribution probability on R, α0, α1 ∈ (−1, 1);β0, β1 ∈ R and

((ε2n, ε2n+1), n ≥ 1

)forms a sequence of

centered i.i.d bivariate random variables with covariance matrix

Γ = σ2(

1 ρρ 1

), σ2 > 0, ρ ∈ (−1, 1).

Page 13: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

The binary tree T

G0 G1 G2 Grn

Page 14: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

The binary tree T

Gq = 2q, 2q + 1, · · · , 2q+1 − 1, Tr =⋃r

q=0 Gq, rn = [log2 n].

G0 G1 G2 Grn

Page 15: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

The model of bifurcating Markov chain

• (S,S)

• P : S × S2 → [0, 1] such that

P(., A) is measurable for all A ∈ S2,P(x , .) is a probability measure on (S2,S2) for all x ∈ S.

• P0(·, ·) = P(·, · ×S), P1(·, ·) = P(·, S× ·) and Q =P0 + P1

2.

• For all f ∈ B(S3),

Pf : x 7→ Pf (x) =

∫S2

f (x , y , z)P(x , dydz). (2)

• ν a probability on (S,S)

Page 16: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

The model of bifurcating Markov chain

• (S,S)

• P : S × S2 → [0, 1] such that

P(., A) is measurable for all A ∈ S2,P(x , .) is a probability measure on (S2,S2) for all x ∈ S.

• P0(·, ·) = P(·, · ×S), P1(·, ·) = P(·, S× ·) and Q =P0 + P1

2.

• For all f ∈ B(S3),

Pf : x 7→ Pf (x) =

∫S2

f (x , y , z)P(x , dydz). (2)

• ν a probability on (S,S)

Page 17: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

The model of bifurcating Markov chain

• (S,S)

• P : S × S2 → [0, 1] such that

P(., A) is measurable for all A ∈ S2,P(x , .) is a probability measure on (S2,S2) for all x ∈ S.

• P0(·, ·) = P(·, · ×S), P1(·, ·) = P(·, S× ·) and Q =P0 + P1

2.

• For all f ∈ B(S3),

Pf : x 7→ Pf (x) =

∫S2

f (x , y , z)P(x , dydz). (2)

• ν a probability on (S,S)

Page 18: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

The model of bifurcating Markov chain

• (S,S)

• P : S × S2 → [0, 1] such that

P(., A) is measurable for all A ∈ S2,P(x , .) is a probability measure on (S2,S2) for all x ∈ S.

• P0(·, ·) = P(·, · ×S), P1(·, ·) = P(·, S× ·) and Q =P0 + P1

2.

• For all f ∈ B(S3),

Pf : x 7→ Pf (x) =

∫S2

f (x , y , z)P(x , dydz). (2)

• ν a probability on (S,S)

Page 19: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

The model of bifurcating Markov chain

• (S,S)

• P : S × S2 → [0, 1] such that

P(., A) is measurable for all A ∈ S2,P(x , .) is a probability measure on (S2,S2) for all x ∈ S.

• P0(·, ·) = P(·, · ×S), P1(·, ·) = P(·, S× ·) and Q =P0 + P1

2.

• For all f ∈ B(S3),

Pf : x 7→ Pf (x) =

∫S2

f (x , y , z)P(x , dydz). (2)

• ν a probability on (S,S)

Page 20: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

The model of bifurcating Markov chain

• (Xn, n ∈ T) : (Ω,F , (Fr , r ∈ N), P) → (S,S)

•(a) Xn is Frn -measurable for all n ∈ T,(b) L(X1) = ν,(c) for all r ∈ N and for all family (fn, n ∈ Gr ) ⊆ Bb(S3)

E

[∏n∈Gr

fn(Xn, X2n, X2n+1)/Fr

]=∏

n∈Gr

Pfn(Xn).

Page 21: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

The model of bifurcating Markov chain

• (Xn, n ∈ T) : (Ω,F , (Fr , r ∈ N), P) → (S,S)

•(a) Xn is Frn -measurable for all n ∈ T,(b) L(X1) = ν,(c) for all r ∈ N and for all family (fn, n ∈ Gr ) ⊆ Bb(S3)

E

[∏n∈Gr

fn(Xn, X2n, X2n+1)/Fr

]=∏

n∈Gr

Pfn(Xn).

Page 22: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Outline

1 Introduction

2 Goals

3 Framework for the results

4 Results

5 Application

Page 23: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

For all i ∈ T, set ∆i = (Xi , X2i , X2i+1).

MTr (f ) =1|Tr |

∑i∈Tr

f (Xi) if f ∈ B(S)

and

MTr (f ) =1|Tr |

∑i∈Tr

f (∆i) if f ∈ B(S3)

Non asymptotic behavior for MTr (f ) (f ∈ B(S)orB(S3))

A moderate deviation principle for MTr (f )b|Tr |

(for f ∈ B(S3) such

that Pf = 0) where

MTr (f ) =∑i∈Tr

f (∆i) if f ∈ B(S3)

Page 24: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

For all i ∈ T, set ∆i = (Xi , X2i , X2i+1).

MTr (f ) =1|Tr |

∑i∈Tr

f (Xi) if f ∈ B(S)

and

MTr (f ) =1|Tr |

∑i∈Tr

f (∆i) if f ∈ B(S3)

Non asymptotic behavior for MTr (f ) (f ∈ B(S)orB(S3))

A moderate deviation principle for MTr (f )b|Tr |

(for f ∈ B(S3) such

that Pf = 0) where

MTr (f ) =∑i∈Tr

f (∆i) if f ∈ B(S3)

Page 25: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

For all i ∈ T, set ∆i = (Xi , X2i , X2i+1).

MTr (f ) =1|Tr |

∑i∈Tr

f (Xi) if f ∈ B(S)

and

MTr (f ) =1|Tr |

∑i∈Tr

f (∆i) if f ∈ B(S3)

Non asymptotic behavior for MTr (f ) (f ∈ B(S)orB(S3))

A moderate deviation principle for MTr (f )b|Tr |

(for f ∈ B(S3) such

that Pf = 0) where

MTr (f ) =∑i∈Tr

f (∆i) if f ∈ B(S3)

Page 26: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Outline

1 Introduction

2 Goals

3 Framework for the resultsFunctional spaceHypothesis

4 Results

5 Application

Page 27: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Functional space

We will work with the subspace F of B(S) which verifies

(i) F contains the constants,

(ii) F 2 ⊂ F ,

(iii) F ⊗ F ⊂ L1(P(x , .)) for all x ∈ S, and P(F ⊗ F ) ⊂ F ,

(iv) there exists a probability µ on (S,S) such that F ⊂ L1(µ)and lim

r→∞Qr f (x) = (µ, f ) for all x ∈ S and f ∈ F ,

(v) for all f ∈ F , there exists g ∈ F such that for all r ∈ N,|Qr f | ≤ g,

(vi) F ⊂ L1(ν)

Page 28: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Hypothesis

Two cases for the results

(H1) Geometric ergodicity of Q: ∀f ∈ F such that (µ, f ) = 0,∃g ∈ F such that ∀r ∈ N and ∀x ∈ S, |Qr f (x)| ≤ αr g(x) forsome α ∈ (0, 1).

(H2) Uniform geometric ergodicity of Q: ∃c > 0 such that

|Qr f (x)| ≤ cαr for some α ∈ (0, 1) and for all x ∈ S,

Page 29: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Hypothesis

Two cases for the results

(H1) Geometric ergodicity of Q: ∀f ∈ F such that (µ, f ) = 0,∃g ∈ F such that ∀r ∈ N and ∀x ∈ S, |Qr f (x)| ≤ αr g(x) forsome α ∈ (0, 1).

(H2) Uniform geometric ergodicity of Q: ∃c > 0 such that

|Qr f (x)| ≤ cαr for some α ∈ (0, 1) and for all x ∈ S,

Page 30: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Hypothesis

Two cases for the results

(H1) Geometric ergodicity of Q: ∀f ∈ F such that (µ, f ) = 0,∃g ∈ F such that ∀r ∈ N and ∀x ∈ S, |Qr f (x)| ≤ αr g(x) forsome α ∈ (0, 1).

(H2) Uniform geometric ergodicity of Q: ∃c > 0 such that

|Qr f (x)| ≤ cαr for some α ∈ (0, 1) and for all x ∈ S,

Page 31: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Outline

1 Introduction

2 Goals

3 Framework for the results

4 Results

5 Application

Page 32: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

• S.Valère. Bitseki Penda, Hacène. Djellout and Arnaud.Guillin. Deviation inequalities, Moderate deviations andsome limit theorems for bifurcating Markov chains withapplication. arXiv:1111.7303 (2011)

Page 33: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Theorem (Deviation inequalities I)

Let f ∈ F such that (µ, f ) = 0. We assume hypothesis (H1).Then for all r ∈ N

P(|MTr (f )| > δ

)≤

c′

δ4

(14

)r+1if α2 < 1

2

c′

δ4 r2(1

4

)r+1if α2 = 1

2

c′

δ4 α4r+4 if α2 > 12

(3)

where the positive constant c′ depends on α and f .

Page 34: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

When f depends on the mother-daughters triangle (∆i)

Theorem (Deviation inequalities II)

We assume that (H1) is fulfilled. Let f ∈ B(S3)

such that Pf andPf 2 exists and belong to F and (µ, Pf ) = 0. Then for all δ > 0and all r ∈ N

P(∣∣MTr (f )

∣∣ > δ)≤

c′

δ2

(12

)r+1if α2 < 1

2 ;

c′

δ2 r(1

2

)r+1if α2 = 1

2 ;

c′

δ2 α2(r+1) if α2 > 12 ,

(4)

where the positive constant c′ depends on f and α.

If Pf = 0,

P(∣∣MTr (f )

∣∣ > δ)≤ c′

δ4

(14

)r+1

(5)

Page 35: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

When f depends on the mother-daughters triangle (∆i)

Theorem (Deviation inequalities II)

We assume that (H1) is fulfilled. Let f ∈ B(S3)

such that Pf andPf 2 exists and belong to F and (µ, Pf ) = 0. Then for all δ > 0and all r ∈ N

P(∣∣MTr (f )

∣∣ > δ)≤

c′

δ2

(12

)r+1if α2 < 1

2 ;

c′

δ2 r(1

2

)r+1if α2 = 1

2 ;

c′

δ2 α2(r+1) if α2 > 12 ,

(4)

where the positive constant c′ depends on f and α.

If Pf = 0,

P(∣∣MTr (f )

∣∣ > δ)≤ c′

δ4

(14

)r+1

(5)

Page 36: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Ideas for the proofs

Markov inequality+ control of second and fourth order moment of MTr (f ) using(H1) and hypothesis (i)-(vi) on F .

• Notice that the dichotomy around the value α2 = 12

naturally appears in the calculus.

Page 37: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Ideas for the proofs

Markov inequality

+ control of second and fourth order moment of MTr (f ) using(H1) and hypothesis (i)-(vi) on F .

• Notice that the dichotomy around the value α2 = 12

naturally appears in the calculus.

Page 38: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Ideas for the proofs

Markov inequality+ control of second and fourth order moment of MTr (f ) using(H1) and hypothesis (i)-(vi) on F .

• Notice that the dichotomy around the value α2 = 12

naturally appears in the calculus.

Page 39: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Ideas for the proofs

Markov inequality+ control of second and fourth order moment of MTr (f ) using(H1) and hypothesis (i)-(vi) on F .

• Notice that the dichotomy around the value α2 = 12

naturally appears in the calculus.

Page 40: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Under the stronger assumption of uniform geometric ergodicityof Q, we have the following more sharp estimations for theabove empirical mean.

Page 41: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Theorem: Exponential probability inequalities

Let f ∈ Bb(S) such that (µ, f ) = 0. Assume that (H2) issatisfied. Then for all δ > 0 we have

P(

MTr (f ) > δ)≤

exp (c′′δ) exp(−c′δ2|Tr |

), ∀r ∈ N, if α < 1

2

exp (2c′δ(r + 1)) exp(−c′δ2|Tr |

), ∀r ∈ N, if α = 1

2

exp(−c′δ2|Tr |

), ∀r > log(δ/c0)

log α , if 12 < α <

√2

2

exp(−c′δ2 |Tr |

r+1

), ∀r > log(c0/δ)

log√

2, if α =

√2

2 ,

exp(−c′δ2

( 1α2

)r+1)

, ∀r > log(δ/c0)log α , if α >

√2

2

(6)

where c0, c′ and c′′ depend on α, ‖f‖∞ and c.

If f ∈ Bb(S3)

such that (µ, Pf ) = 0 then we have the sameconclusions for MTr (f ).

Page 42: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Theorem: Exponential probability inequalities

Let f ∈ Bb(S) such that (µ, f ) = 0. Assume that (H2) issatisfied. Then for all δ > 0 we have

P(

MTr (f ) > δ)≤

exp (c′′δ) exp(−c′δ2|Tr |

), ∀r ∈ N, if α < 1

2

exp (2c′δ(r + 1)) exp(−c′δ2|Tr |

), ∀r ∈ N, if α = 1

2

exp(−c′δ2|Tr |

), ∀r > log(δ/c0)

log α , if 12 < α <

√2

2

exp(−c′δ2 |Tr |

r+1

), ∀r > log(c0/δ)

log√

2, if α =

√2

2 ,

exp(−c′δ2

( 1α2

)r+1)

, ∀r > log(δ/c0)log α , if α >

√2

2

(6)

where c0, c′ and c′′ depend on α, ‖f‖∞ and c.If f ∈ Bb

(S3)

such that (µ, Pf ) = 0 then we have the sameconclusions for MTr (f ).

Page 43: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

ideas for the proof

Chernoff inequality + successive conditioning and successiveapplications of Azuma-Bennet-Hoeffding using (H2).

• Once again, notice that the dichotomy around α = 12 and

α2 = 12 in (6) naturally appears from the calculations.

Page 44: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

ideas for the proof

Chernoff inequality

+ successive conditioning and successiveapplications of Azuma-Bennet-Hoeffding using (H2).

• Once again, notice that the dichotomy around α = 12 and

α2 = 12 in (6) naturally appears from the calculations.

Page 45: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

ideas for the proof

Chernoff inequality + successive conditioning and successiveapplications of Azuma-Bennet-Hoeffding using (H2).

• Once again, notice that the dichotomy around α = 12 and

α2 = 12 in (6) naturally appears from the calculations.

Page 46: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

ideas for the proof

Chernoff inequality + successive conditioning and successiveapplications of Azuma-Bennet-Hoeffding using (H2).

• Once again, notice that the dichotomy around α = 12 and

α2 = 12 in (6) naturally appears from the calculations.

Page 47: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Moderate deviation principle for MTr (f )

Let (bn) be an increasing sequence of positive real numberssuch that

(I) bn√n−→ +∞,

(II) if α2 < 12 , the sequence (bn) is such that

bn

n−→ 0,

(III) if α2 = 12 , the sequence (bn) is such that

bn log nn

−→ 0,

(IV) if α2 > 12 , the sequence (bn) is such that

bnαrn+1

√n

−→ 0.

• The conditions (II)-(IV) come from deviation inequalities(6).

Page 48: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Moderate deviation principle for MTr (f )

Let (bn) be an increasing sequence of positive real numberssuch that

(I) bn√n−→ +∞,

(II) if α2 < 12 , the sequence (bn) is such that

bn

n−→ 0,

(III) if α2 = 12 , the sequence (bn) is such that

bn log nn

−→ 0,

(IV) if α2 > 12 , the sequence (bn) is such that

bnαrn+1

√n

−→ 0.

• The conditions (II)-(IV) come from deviation inequalities(6).

Page 49: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Moderate deviation principle for MTr (f )

Let (bn) be an increasing sequence of positive real numberssuch that

(I) bn√n−→ +∞,

(II) if α2 < 12 , the sequence (bn) is such that

bn

n−→ 0,

(III) if α2 = 12 , the sequence (bn) is such that

bn log nn

−→ 0,

(IV) if α2 > 12 , the sequence (bn) is such that

bnαrn+1

√n

−→ 0.

• The conditions (II)-(IV) come from deviation inequalities(6).

Page 50: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Moderate deviation principle for MTr (f )

Let f ∈ Bb(S3) such that Pf = 0. (Zr ) =(

1b|Tr |

MTr (f ))

Theorem (Moderate deviation principle)

Assume that (H2) is satisfied. Let (bn) be a sequence of realnumbers satisfying the above assumptions (I)-(IV), then (Zr )

satisfies a MDP in R with the speedb2|Tr ||Tr | and rate function

I(x) = x2

2(µ,Pf 2).

Particularly, for all δ > 0, we have

limr→∞

|Tr |b2|Tr |

log P(

1b|Tr |

|M|Tr |(f )| > δ

)= −I(δ). (7)

P(

1bTr

|MTr (f )| ≥ δ

)∼ exp

(−

b2Tr

Tr

δ2

2(µ, Pf 2)

)(7′)

Page 51: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Moderate deviation principle for MTr (f )

Let f ∈ Bb(S3) such that Pf = 0. (Zr ) =(

1b|Tr |

MTr (f ))

Theorem (Moderate deviation principle)

Assume that (H2) is satisfied. Let (bn) be a sequence of realnumbers satisfying the above assumptions (I)-(IV), then (Zr )

satisfies a MDP in R with the speedb2|Tr ||Tr | and rate function

I(x) = x2

2(µ,Pf 2).

Particularly, for all δ > 0, we have

limr→∞

|Tr |b2|Tr |

log P(

1b|Tr |

|M|Tr |(f )| > δ

)= −I(δ). (7)

P(

1bTr

|MTr (f )| ≥ δ

)∼ exp

(−

b2Tr

Tr

δ2

2(µ, Pf 2)

)(7′)

Page 52: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Moderate deviation principle for MTr (f )

Let f ∈ Bb(S3) such that Pf = 0. (Zr ) =(

1b|Tr |

MTr (f ))

Theorem (Moderate deviation principle)

Assume that (H2) is satisfied. Let (bn) be a sequence of realnumbers satisfying the above assumptions (I)-(IV), then (Zr )

satisfies a MDP in R with the speedb2|Tr ||Tr | and rate function

I(x) = x2

2(µ,Pf 2).

Particularly, for all δ > 0, we have

limr→∞

|Tr |b2|Tr |

log P(

1b|Tr |

|M|Tr |(f )| > δ

)= −I(δ). (7)

P(

1bTr

|MTr (f )| ≥ δ

)∼ exp

(−

b2Tr

Tr

δ2

2(µ, Pf 2)

)(7′)

Page 53: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

Outline

1 Introduction

2 Goals

3 Framework for the results

4 Results

5 Application

Page 54: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

We consider the BAR(1) process

L(X1) = ν, and∀n ≥ 1,

X2n = α0Xn + β0 + ε2n

X2n+1 = α1Xn + β1 + ε2n+1,

The noise values in a compact set.F = Cb(R)(H2) are automatically satisfied with α = max(|α0|, |α1|).The least square estimator θr =

(αr

0, βr0, α

r1, β

r1

)of

θ = (α0, β0, α1, β1) is given by, for η ∈ 0, 1

αrη =

|Tr |−1 ∑i∈Tr

Xi X2i+η−(|Tr |−1 ∑

i∈Tr

Xi

)(|Tr |−1 ∑

i∈Tr

X2i+η

)

|Tr |−1∑

i∈Tr

X 2i −(|Tr |−1

∑i∈Tr

Xi

)2

βrη = |Tr |−1 ∑

i∈Tr

X2i+η − αrη|Tr |−1 ∑

i∈Tr

Xi .

(8)

Page 55: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

We consider the BAR(1) process

L(X1) = ν, and∀n ≥ 1,

X2n = α0Xn + β0 + ε2n

X2n+1 = α1Xn + β1 + ε2n+1,

The noise values in a compact set.

F = Cb(R)(H2) are automatically satisfied with α = max(|α0|, |α1|).The least square estimator θr =

(αr

0, βr0, α

r1, β

r1

)of

θ = (α0, β0, α1, β1) is given by, for η ∈ 0, 1

αrη =

|Tr |−1 ∑i∈Tr

Xi X2i+η−(|Tr |−1 ∑

i∈Tr

Xi

)(|Tr |−1 ∑

i∈Tr

X2i+η

)

|Tr |−1∑

i∈Tr

X 2i −(|Tr |−1

∑i∈Tr

Xi

)2

βrη = |Tr |−1 ∑

i∈Tr

X2i+η − αrη|Tr |−1 ∑

i∈Tr

Xi .

(8)

Page 56: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

We consider the BAR(1) process

L(X1) = ν, and∀n ≥ 1,

X2n = α0Xn + β0 + ε2n

X2n+1 = α1Xn + β1 + ε2n+1,

The noise values in a compact set.F = Cb(R)

(H2) are automatically satisfied with α = max(|α0|, |α1|).The least square estimator θr =

(αr

0, βr0, α

r1, β

r1

)of

θ = (α0, β0, α1, β1) is given by, for η ∈ 0, 1

αrη =

|Tr |−1 ∑i∈Tr

Xi X2i+η−(|Tr |−1 ∑

i∈Tr

Xi

)(|Tr |−1 ∑

i∈Tr

X2i+η

)

|Tr |−1∑

i∈Tr

X 2i −(|Tr |−1

∑i∈Tr

Xi

)2

βrη = |Tr |−1 ∑

i∈Tr

X2i+η − αrη|Tr |−1 ∑

i∈Tr

Xi .

(8)

Page 57: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

We consider the BAR(1) process

L(X1) = ν, and∀n ≥ 1,

X2n = α0Xn + β0 + ε2n

X2n+1 = α1Xn + β1 + ε2n+1,

The noise values in a compact set.F = Cb(R)(H2) are automatically satisfied with α = max(|α0|, |α1|).

The least square estimator θr =(αr

0, βr0, α

r1, β

r1

)of

θ = (α0, β0, α1, β1) is given by, for η ∈ 0, 1

αrη =

|Tr |−1 ∑i∈Tr

Xi X2i+η−(|Tr |−1 ∑

i∈Tr

Xi

)(|Tr |−1 ∑

i∈Tr

X2i+η

)

|Tr |−1∑

i∈Tr

X 2i −(|Tr |−1

∑i∈Tr

Xi

)2

βrη = |Tr |−1 ∑

i∈Tr

X2i+η − αrη|Tr |−1 ∑

i∈Tr

Xi .

(8)

Page 58: Deviation inequalities and moderate deviation principle ...reynaudb/jps/ergolim/BPSV_CIRM_2012.pdf · Université Blaise Pascal ... Figure: Cell division from E. J. Stewart and al.

Introduction Goals Framework for the results Results Application

We consider the BAR(1) process

L(X1) = ν, and∀n ≥ 1,

X2n = α0Xn + β0 + ε2n

X2n+1 = α1Xn + β1 + ε2n+1,

The noise values in a compact set.F = Cb(R)(H2) are automatically satisfied with α = max(|α0|, |α1|).The least square estimator θr =

(αr

0, βr0, α

r1, β

r1

)of

θ = (α0, β0, α1, β1) is given by, for η ∈ 0, 1

αrη =

|Tr |−1 ∑i∈Tr

Xi X2i+η−(|Tr |−1 ∑

i∈Tr

Xi

)(|Tr |−1 ∑

i∈Tr

X2i+η

)

|Tr |−1∑

i∈Tr

X 2i −(|Tr |−1

∑i∈Tr

Xi

)2

βrη = |Tr |−1 ∑

i∈Tr

X2i+η − αrη|Tr |−1 ∑

i∈Tr

Xi .

(8)

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Introduction Goals Framework for the results Results Application

µ1 : Θ → R and µ2 : Θ× R∗+ → R by writing

(µ, x) = µ1(θ) and (µ, x2) = µ2(θ, σ2), (9)

where θ = (α0, β0, α1, β1) ∈ Θ = (−1, 1)× R× (−1, 1)× R, andµ is the stationary distribution of Q.

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Theorem

∀δ > 0 and ∀γ < min(

c1b1+δ , c1b

1+√

δ, c1b

1+ 4√δ

), where c1 = c1(µ1, µ2)

we have P(∥∥∥θr − θ

∥∥∥ > δ)≤

exp(c′′(γδ)1−p/2

)exp

(−c′(γδ)2−p|Tr |

), ∀r ∈ N, if α < 1

2

exp(c′(γδ(r + 1))1−p/2

)exp

(−c′(γδ)2−p|Tr |

), ∀r ∈ N, if α = 1

2

exp(−c′(γδ)2−p|Tr |

), ∀r >

log((γδ)1−p/2/c0)log α , if 1

2 < α <√

22

exp(−c′(γδ)2−p |Tr |

r+1

), ∀r >

log(c0/(γδ)1−p/2)log

√2

, if α =√

22

exp(−c′(γδ)2−p

( 1α2

)r+1)

, ∀r >log((γδ)1−p/2/c0)

log α , if α >√

22 ,

(10)

where c′ and c′′ depend on α, ‖f‖∞ and c, c0 depends on α,‖f‖∞, c and γ, and p ∈ 0, 1, 3/2.

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Introduction Goals Framework for the results Results Application

End.