Soutenance d’Habilitation à Diriger des Recherches ————————– Théorie spatiale des extrêmes et propriétés des processus max-stables C LÉMENT D OMBRY Laboratoire de Mathématiques et Applications, Université de Poitiers. Poitiers, le 8 Novembre 2012 C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 1 / 39
Overview of the research interests Structure of the talk Overview of the research interests 1 Conditional distribution of max-i.d. random fields 2 Strong mixing properties of max-i.d. processes 3 Intermediate regime for aggregated ON/OFF sources 4 A stochastic gradient algorithm for the p-means 5 C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 2 / 39
Overview of the research interests Theme 1 : Stochastic models in biology, geometry . . . PhD dissertation : "Applications of large deviation theory". Supervision by C.Mazza and N.Guillotin-Plantard. Asymptotics of stochastic models from biology and informatics. Interest in discrete stochastic models arising from various domains : Biology and population dynamics : ⊲ A stochastic model for DNA denaturation. ⊲ Asymptotic study of a mutation/selection genetic algorithm. ⊲ Phenotypic diversity and population growth in a fluctuating environment (with V.Bansaye and C.Mazza). Statistical Physics : ⊲ The Curie-Weiss model with quasiperiodic external random field (with N.Guillotin). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 3 / 39
Overview of the research interests Theme 1 : Stochastic models in biology, geometry . . . Informatic and stochastic algorithms : ⊲ Data structures with dynamical random transitions (with R.Schott and N.Guillotin). ⊲ The stochastic k -server problem on the circle (with E.Upfal and N.Guillotin). Geometry : ⊲ Betti numbers of random polygon surfaces (with C.Mazza). ⊲ Stochastic gradient algorithm for the p -mean on a manifold (with M.Arnaudon, A.Phan and L.Yang). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 4 / 39
Overview of the research interests Theme 2 : Limit theorems, heavy tails and LRD Interest in (functional) limit theorems for stochastic models in presence of heavy tails and/or long range dependence (LRD). Asymptotic theory is often very rich with different regimes and nice limit processes such as : - fractional Brownian motion, - self-similar non-Gaussian stable processes, - fractional Poisson process, - telecom process . . . Some motivations from telecommunication models (Internet traffic, transmission network . . . ) C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 5 / 39
Overview of the research interests Theme 2 : Limit theorems, heavy tails and LRD Particular models investigated : ⊲ Random walks in random sceneries and random reward schemas (with N.Guillotin and S.Cohen). ⊲ Weighted random ball models (with J.-C. Breton). ⊲ Aggregation of sources based on ON/OFF or renewal processes (with I.Kaj). ⊲ Le Page series in Skohorod space (with Y.Davydov). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 6 / 39
Overview of the research interests Theme 3 : Spatial EVT and max-stable processes Interest in spatial extreme value theory (EVT) and max-stable random fields. Deep connections with the previous theme via : - the theory of regular variation, - a parallel between sum-stable and max-stable processes, - importance of limit theorems. Some motivations from models in environmental science (heat waves, flood, storm . . . ) Supervision of the PhD thesis of Frédéric Eyi-Minko. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 7 / 39
Overview of the research interests Theme 3 : Spatial EVT and max-stable processes Results obtained : ⊲ Properties and asymptotics of extremal shot noises. ⊲ A point process approach for the maxima of i.i.d. random fields (with F .Eyi-Minko). ⊲ Conditional distribution of max-i.d. random fields (with F .Eyi-Minko). ⊲ Conditional simulation of max-stable processes (with M.Ribatet and F .Eyi-Minko). ⊲ Strong mixing properties of continuous max-i.d. random fields (with F .Eyi-Minko). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 8 / 39
Conditional distribution of max-i.d. random fields Structure of the talk Overview of the research interests 1 Conditional distribution of max-i.d. random fields 2 Strong mixing properties of max-i.d. processes 3 Intermediate regime for aggregated ON/OFF sources 4 A stochastic gradient algorithm for the p-means 5 C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 9 / 39
Conditional distribution of max-i.d. random fields Motivations Needs for modeling extremes in environmental sciences : - maximal temperatures in a heat wave, - intensity of winds during a storm, - water heights in a flood ... Spatial extreme value theory - geostatistics of extremes : ⊲ Schlather (’02), Models for stationary max-stable random fields. ⊲ de Haan & Pereira (’06), Spatial extremes : Models for the stationary case. ⊲ Davison, Ribatet & Padoan (’11), Statistical modelling of spatial extremes. Max-stable random fields play a crucial role as possible limits of normalized pointwise maxima of i.i.d. random fields. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 10 / 39
Conditional distribution of max-i.d. random fields Motivations Observations of a max-stable process η at some stations only : i = 1 , . . . , k . (O) η ( s i ) = y i , How to predict what happens at other locations ? We are naturally lead to consider the conditional distribution of η given the observations (O). Different goals : - theoretical formulas for the conditional distribution, - sample from the conditional distribution, - compute (numerically) the conditional median or quantiles ... Results for spectrally discrete max-stable processes : ⊲ Wang & Stoev (’11), Conditional sampling of spectrally discrete max-stable processes. New results for max-stable and even max-i.d. processes. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 11 / 39
Conditional distribution of max-i.d. random fields Structure of max-i.d. processes Theorem (de Haan ’84, Giné Hahn & Vatan ’90) For any continuous, max-i.d. random process η = ( η ( t )) t ∈ T satisfying ess inf η ( t ) ≡ 0, there exists a unique Borel measure µ on C 0 = C ( T , [ 0 , + ∞ )) \ { 0 } such that � � � � � L with Φ ∼ PPP ( µ ) . η ( t ) = φ ( t ) t ∈ T , t ∈ T φ ∈ Φ F IGURE : A realization of Φ and η = max (Φ) for Smith 1D storm process. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 12 / 39
Conditional distribution of max-i.d. random fields Hitting scenario and extremal functions Observations { η ( s i ) = y i , 1 ≤ i ≤ k } with η ( s ) = � φ ∈ Φ φ ( s ) . Assume that the law of η ( s i ) has no atom, 1 ≤ i ≤ k . Then, with probability 1, ∃ ! φ i ∈ Φ , φ i ( s i ) = η ( s i ) . Definition of the following random objects : the hitting scenario Θ , a partition of S = { s 1 , . . . , s k } with ℓ blocks, the extremal functions ϕ + 1 , · · · , ϕ + ℓ ∈ Φ , the subextremal functions Φ − S ⊂ Φ . Example with k = 4 : Θ = ( { s 1 } , { s 2 } , { s 3 , s 4 } ) Θ = ( { s 1 , s 2 } , { s 3 , s 4 } ) C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 13 / 39
Conditional distribution of max-i.d. random fields Joint distribution Let P k be the set of partitions of { s 1 , · · · , s k } . We note s = ( s 1 , . . . , s k ) . Theorem For τ = ( τ 1 , . . . , τ ℓ ) ∈ P k , A ⊂ C ℓ 0 and B ⊂ M p ( C 0 ) measurable � � Θ = τ, ( ϕ + 1 , . . . , ϕ + ℓ ) ∈ A , Φ − S ∈ B P � 1 {∀ j ∈ [ ] , f j > τ j ∨ j ′� = j f j ′ } 1 { ( f 1 , ··· , f ℓ ) ∈ A } = [ 1 ,ℓ ] C ℓ 0 P [ { Φ ∈ B } ∩ {∀ φ ∈ Φ , φ < S ∨ ℓ j = 1 f j } ] µ ( df 1 ) · · · µ ( df ℓ ) Furthermore, the law ν s of η ( s ) is equal to ν s = � τ ∈P k ν τ s with ℓ � � � � � ν τ exp s ( d y ) = − µ ( f ( s ) � < y ) µ f ( s τ c j ) < y τ c j , f ( s τ j ) ∈ d y τ j . j = 1 C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 14 / 39
Conditional distribution of max-i.d. random fields Conditional distribution A three step procedure for the conditional law of η given η ( s ) = y : Step 1 : sample Θ from the conditional law w.r.t. η ( s ) = y . C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 15 / 39
Conditional distribution of max-i.d. random fields Conditional distribution Step 2 : sample ( ϕ + j ) from the conditional law w.r.t. η ( s ) = y , Θ = τ . Step 3 : sample Φ − s from the conditional law w.r.t. η ( s ) = y , Θ = τ , ( ϕ + j ) = ( f j ) . Finally, set η ( t ) = � | Θ | j ( t ) � j = 1 ϕ + s φ ( t ) . φ ∈ Φ − C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 16 / 39
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