Deviation inequalities and moderate deviation principle for Bifurcating Markov

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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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Outline

1 Introduction Motivation

The model of bifurcating Markov chain

2 Goals

3 Framework for the results

4 Results

5 Application

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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)

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

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Motivation

Guyon, J. Bize, A. Paul, G. Stewart, E.J. Delmas, J.F.

Taddéi, F. Statistical study of cellular aging. CEMRACS 2004 Proceedings, ESAIM Proceedings, (2005), 14, pp.

100-114.

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Motivation

First order bifurcating autoregressive process BAR(1)

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Motivation

Xⁿ

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Motivation

L(X₁) =ν, and ∀n≥1,







X_2n =α₀Xn+β₀+ε_2n

X_2n+1=α₁X_n+β₁+ε_2n+1, (1)

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Motivation

whereν is a distribution probability onR,α₀, α₁ ∈(−1,1);

β₀, β₁ ∈ R and (ε_2n, ε_2n+1),n ≥ 1

forms a sequence of centered i.i.d bivariate random variables with covariance matrix

Γ =σ² 1 ρ

ρ 1

, σ²>0, ρ∈(−1,1).

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Motivation

θ= (α₀, β₀, α₁, β₁), σ andρ.

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

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Motivation

θ= (α₀, β₀, α₁, β₁), σ andρ.

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

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Motivation

L(X₁) =ν, and ∀n≥1,







X_2n+1=α1X_n+β1+ε2n+1, (1)

whereνis a distribution probability onR,α₀, α₁∈(−1,1);

β₀, β₁∈Rand (ε_2n, ε_2n+1),n≥1

forms a sequence of

centered i.i.d bivariate random variables with covariance matrix Γ =σ²

1 ρ

ρ 1

, σ²>0, ρ∈(−1,1).

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The binary treeT

G0 G1 G2 Grn

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The binary treeT

Gq ={2^q,2^q+1,· · · ,2^q+1−1},Tr =Sr

q=0Gq,rn= [log₂n].

G0 G1 G2 Grn

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• (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) =

Z

S²

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

• ν a probability on(S,S)

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• (S,S)

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

2 .

Z

S²

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

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• (S,S)

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

2 .

Z

S²

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

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• (S,S)

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

2 .

Z

S²

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

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• (S,S)

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

2 .

Z

S²

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

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• (X_n,n∈T) : (Ω,F,(F_r,r ∈N),P)→(S,S)

•

(a) XnisF_r_n-measurable for all n∈T, (b) L(X₁) =ν,

(c) for all r ∈Nand for all family(f_n,n∈Gr)⊆ B_b(S³)

E

" Y

n∈Gr

fn(Xn,X2n,X2n+1). F_r

#

= Y

n∈Gr

Pfn(Xn).

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• (X_n,n∈T) : (Ω,F,(F_r,r ∈N),P)→(S,S)

•

(a) XnisF_r_n-measurable for all n∈T, (b) L(X₁) =ν,

(c) for all r ∈Nand for all family(f_n,n∈Gr)⊆ B_b(S³)

E

"

Y

n∈Gr

fn(Xn,X2n,X2n+1). F_r

#

= Y

n∈Gr

Pfn(Xn).

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Outline

1 Introduction

2 Goals

4 Results

5 Application

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For all i ∈T, set∆_i = (X_i,X_2i,X_2i₊₁).

M_T_r(f) = 1

|Tr| X

i∈Tr

f(X_i) if f ∈ B(S) and

M_T_r(f) = 1

|Tr| X

i∈Tr

f(∆_i) if f ∈ B(S³)

Non asymptotic behavior for M_T_r(f)(f ∈ B(S)orB(S³)) A moderate deviation principle for ^M_b^Tr^(f⁾

|Tr| (for f ∈ B(S³)such that Pf =0) where

M_T_r(f) =X

i∈Tr

f(∆_i) if f ∈ B(S³)

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M_T_r(f) = 1

|Tr| X

i∈Tr

M_T_r(f) = 1

|Tr| X

i∈Tr

f(∆_i) if f ∈ B(S³) Non asymptotic behavior for M_T_r(f)(f ∈ B(S)orB(S³))

A moderate deviation principle for ^M_b^Tr^(f⁾

M_T_r(f) =X

i∈Tr

f(∆_i) if f ∈ B(S³)

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M_T_r(f) = 1

|Tr| X

i∈Tr

M_T_r(f) = 1

|Tr| X

i∈Tr

f(∆_i) if f ∈ B(S³)

Non asymptotic behavior for M_T_r(f)(f ∈ B(S)orB(S³)) A moderate deviation principle for ^M_b^Tr^(f⁾

M_T_r(f) =X

i∈Tr

f(∆_i) if f ∈ B(S³)

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Outline

1 Introduction

2 Goals

3 Framework for the results Functional space

Hypothesis

4 Results

5 Application

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Functional space

We will work with the subspace F ofB(S)which verifies (i) F contains the constants,

(ii) F²⊂F ,

(iii) F ⊗F ⊂L¹(P(x, .))for all x ∈S, and P(F ⊗F)⊂F , (iv) there exists a probabilityµon(S,S)such that F ⊂L¹(µ)

and lim

r→∞Q^rf(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,

|Q^rf| ≤g, (vi) F ⊂L¹(ν)

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Hypothesis

Two cases for the results

(H1) Geometric ergodicity of Q: ∀f ∈F such that(µ,f) =0,

∃g∈F such that∀r ∈Nand∀x ∈S,|Q^rf(x)| ≤α^rg(x)for someα∈(0,1).

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

|Q^rf(x)| ≤cα^r for some α∈(0,1) and for all x ∈S,

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Hypothesis

(H1) Geometric ergodicity of Q:∀f ∈F such that(µ,f) =0,

|Q^rf(x)| ≤cα^r for some α∈(0,1) and for all x ∈S,

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Hypothesis

(H1) Geometric ergodicity of Q:∀f ∈F such that(µ,f) =0,

|Q^rf(x)| ≤cα^r for some α ∈(0,1) and for all x ∈S,

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Outline

1 Introduction

2 Goals

4 Results

5 Application

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• S.Valère. Bitseki Penda, Hacène. Djellout and Arnaud.

Guillin. Deviation inequalities, Moderate deviations and some limit theorems for bifurcating Markov chains with application. arXiv:1111.7303 (2011)

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Theorem (Deviation inequalities I)

Let f ∈F such that(µ,f) =0.We assume hypothesis (H1).

Then for all r ∈N

P

|M_T_r(f)|> δ

≤











c⁰ δ⁴

1 4

r+1

if α²< ¹₂

c⁰

δ⁴r² ¹₄r+1

if α²= ¹₂

c⁰

δ⁴α^4r+4 if α²> ¹₂

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where the positive constant c⁰ depends onαand f .

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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 and Pf²exists and belong to F and(µ,Pf) =0. Then for allδ >0 and all r ∈N

P

M_T_r(f) > δ

≤











c⁰ δ²

1 2

r+1

if α²< ¹₂;

c⁰

δ²r ¹₂r+1

if α²= ¹₂;

c⁰

δ²α^2(r⁺¹⁾ if α²> ¹₂,

(4)

where the positive constant c⁰ depends on f andα.

If Pf =0, P

M_T_r(f) > δ

≤ c⁰ δ⁴

1 4

r+1

(5)

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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 and Pf²exists and belong to F and(µ,Pf) =0. Then for allδ >0 and all r ∈N

P

M_T_r(f) > δ

≤











c⁰ δ²

1 2

r+1

if α²< ¹₂;

c⁰

δ²r ¹₂r+1

if α²= ¹₂;

c⁰

δ²α^2(r⁺¹⁾ if α²> ¹₂,

(4)

where the positive constant c⁰ depends on f andα.

If Pf =0, P

M_T_r(f) > δ

≤ c⁰ δ⁴

1 4

r+1

(5)

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Ideas for the proofs

Markov inequality

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

• Notice that the dichotomy around the valueα²= ¹₂ naturally appears in the calculus.

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Markov inequality

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Markov inequality

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Markov inequality

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

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Theorem: Exponential probability inequalities

Let f ∈ B_b(S)such that(µ,f) =0. Assume that (H2) is satisfied. Then for allδ >0 we have

P

M_T_r(f)> δ

≤











exp(c⁰⁰δ)exp −c⁰δ²|Tr|

,∀r ∈N,ifα < ¹₂ exp(2c⁰δ(r +1))exp −c⁰δ²|Tr|

,∀r ∈N,ifα= ¹₂ exp −c⁰δ²|Tr|

,∀r > ^log(δ/c_log_α⁰⁾,if ¹₂ < α <

√2 2

exp

−c⁰δ²_r^|^T₊₁^r^|

,∀r > ^log(c⁰^/δ)

log√

2 ,ifα=

√ 2 2 ,

exp

−c⁰δ² _α¹2

r+1

,∀r > ^log(δ/c_log_α⁰⁾,ifα >

√ 2 2

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where c₀,c⁰ and c⁰⁰depend onα,kfk_∞and c.

If f ∈ B_b S³

such that(µ,Pf) =0 then we have the same conclusions for M_T_r(f).

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Theorem: Exponential probability inequalities

Let f ∈ B_b(S)such that(µ,f) =0. Assume that (H2) is satisfied. Then for allδ >0 we have

P

M_T_r(f)> δ

≤











exp(c⁰⁰δ)exp −c⁰δ²|Tr|

,∀r ∈N,ifα < ¹₂ exp(2c⁰δ(r +1))exp −c⁰δ²|Tr|

,∀r ∈N,ifα= ¹₂ exp −c⁰δ²|Tr|

,∀r > ^log(δ/c_log_α⁰⁾,if ¹₂ < α <

√2 2

exp

−c⁰δ²_r^|^T₊₁^r^|

,∀r > ^log(c⁰^/δ)

log√

2 ,ifα=

√ 2 2 ,

exp

−c⁰δ² _α¹2

r+1

,∀r > ^log(δ/c_log_α⁰⁾,ifα >

√ 2 2

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where c₀,c⁰ and c⁰⁰depend onα,kfk_∞and c.

If f ∈ B_b S³

such that(µ,Pf) =0 then we have the same conclusions for M_T_r(f).

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ideas for the proof

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

• Once again, notice that the dichotomy aroundα= ¹₂ and α²= ¹₂ in (6) naturally appears from the calculations.

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Chernoff inequality

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

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Moderate deviation principle for M_T_r(f)

Let(b_n)be an increasing sequence of positive real numbers such that

(I) ^√^bⁿ

n −→+∞,

(II) ifα²< ¹₂, the sequence(bn)is such that bn

n −→0, (III) ifα²= ¹₂, the sequence(bn)is such that b_nlog n

n −→0, (IV) ifα²> ¹₂, the sequence(b_n)is such that b_nα^rⁿ⁺¹

√n −→0.

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

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(I) ^√^bⁿ_n −→+∞,

√n −→0.

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(I) ^√^bⁿ_n −→+∞,

√n −→0.

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Moderate deviation principle for M_T_r(f) Let f ∈ B_b(S³)such that Pf =0.(Z_r) =

1

b|Tr|M_T_r(f)

Theorem (Moderate deviation principle)

Assume that (H2) is satisfied. Let(b_n)be a sequence of real numbers satisfying the above assumptions (I)-(IV), then(Zr) satisfies a MDP inRwith the speed ^b

2

|Tr|

|T^r| and rate function I(x) = _2(µ,Pf^x² 2).

Particularly, for allδ >0,we have

r→∞lim

|Tr| b_|²

Tr|

logP 1

b_|_T_r_||M_|_T_r_|(f)|> δ

=−I(δ). (7)

P 1

b_T_r|M_T_r_(f₎| ≥δ

∼exp −b²

Tr

δ² 2(µ,Pf²)

!

(7⁰)

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1

b|Tr|M_T_r(f) Theorem (Moderate deviation principle)

2

|Tr|

r→∞lim

|Tr| b_|²

Tr|

logP 1

b_|_T_r_||M_|_T_r_|(f)|> δ

=−I(δ). (7)

P 1

∼exp −b²

Tr

δ² 2(µ,Pf²)

!

(7⁰)

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1

b|Tr|M_T_r(f) Theorem (Moderate deviation principle)

2

|Tr|

r→∞lim

|Tr| b_|²

Tr|

logP 1

b_|_T_r_||M_|_T_r_|(f)|> δ

=−I(δ). (7)

P 1

∼exp −b²

Tr

δ² 2(µ,Pf²)

!

(7⁰)

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Outline

1 Introduction

2 Goals

4 Results

5 Application

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We consider the BAR(1) process L(X₁) =ν,and∀n≥1,







X_2n+1=α1X_n+β1+ε2n+1,

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

(H2) are automatically satisfied withα=max(|α₀|,|α₁|). The least square estimatorθˆ^r = ˆα^r₀,βˆ₀^r,αˆ^r₁,βˆ₁^r

of θ= (α0, β0, α1, β1)is given by, forη∈ {0,1}









 ˆ α^r_η =

|Tr|⁻¹ P

i∈Tr

X_iX_2i+η− |Tr|⁻¹ P

i∈Tr

X_i

!

|Tr|⁻¹ P

i∈Tr

X_2i+η

!

|Tr|⁻¹ P

i∈Tr

X_i²− |Tr|⁻¹ P

i∈Tr

Xi

!2

βˆ_η^r =|Tr|⁻¹ P

i∈Tr

X_2i+η−αˆ^r_η|Tr|⁻¹ P

i∈Tr

X_i.

(8)

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





X_2n+1=α1X_n+β1+ε2n+1, The noise values in a compact set.

F =C_b(R)









 ˆ α^r_η =

|Tr|⁻¹ P

i∈Tr

X_i

!

|Tr|⁻¹ P

i∈Tr

X_2i+η

!

|Tr|⁻¹ P

i∈Tr

X_i²− |Tr|⁻¹ P

i∈Tr

Xi

!2

i∈Tr

X_i.

(8)

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





F =C_b(R)









 ˆ α^r_η =

|Tr|⁻¹ P

i∈Tr

X_i

!

|Tr|⁻¹ P

i∈Tr

X_2i+η

!

|Tr|⁻¹ P

i∈Tr

X_i²− |Tr|⁻¹ P

i∈Tr

Xi

!2

i∈Tr

X_i.

(8)

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





F =C_b(R)

(H2) are automatically satisfied withα=max(|α₀|,|α₁|).

The least square estimatorθˆ^r = ˆα^r₀,βˆ₀^r,αˆ^r₁,βˆ₁^r of θ= (α0, β0, α1, β1)is given by, forη∈ {0,1}









 ˆ α^r_η =

|Tr|⁻¹ P

i∈Tr

X_i

!

|Tr|⁻¹ P

i∈Tr

X_2i+η

!

|Tr|⁻¹ P

i∈Tr

X_i²− |Tr|⁻¹ P

i∈Tr

Xi

!2

i∈Tr

X_i.

(8)

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





F =C_b(R)

(H2) are automatically satisfied withα=max(|α₀|,|α₁|).

The least square estimatorθˆ^r = ˆα^r₀,βˆ₀^r,αˆ^r₁,βˆ₁^r of θ= (α0, β0, α1, β1)is given by, forη∈ {0,1}









 ˆ α^r_η =

|Tr|⁻¹ P

i∈Tr

X_i

!

|Tr|⁻¹ P

i∈Tr

X_2i+η

!

|T^r|⁻¹ P

i∈Tr

X_i²− |T^r|⁻¹ P

i∈Tr

Xi

!2

i∈Tr

X_i.

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