Soutenance dHabilitation Diriger des Recherches Thorie - - PowerPoint PPT Presentation

Soutenance d’Habilitation à Diriger des Recherches ————————– Théorie spatiale des extrêmes et propriétés des processus max-stables

CLÉMENT DOMBRY

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

1

Overview of the research interests

2

Conditional distribution of max-i.d. random fields

3

Strong mixing properties of max-i.d. processes

4

Intermediate regime for aggregated ON/OFF sources

5

A stochastic gradient algorithm for the p-means

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

1

Overview of the research interests

2

Conditional distribution of max-i.d. random fields

3

Strong mixing properties of max-i.d. processes

4

Intermediate regime for aggregated ON/OFF sources

5

A stochastic gradient algorithm for the p-means

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 : η(si) = yi, i = 1, . . . , k. (O) 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 C0 = C(T, [0, +∞)) \ {0} such that

• η(t)
• t∈T

L

=

φ∈Φ

φ(t)

• t∈T,

with Φ ∼ PPP(µ).

FIGURE: 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(si) = yi, 1 ≤ i ≤ k} with η(s) =

φ∈Φ φ(s).

Assume that the law of η(si) has no atom, 1 ≤ i ≤ k. Then, with probability 1, ∃! φi ∈ Φ, φi(si) = η(si). Definition of the following random objects :

the hitting scenario Θ, a partition of S = {s1, . . . , sk} with ℓ blocks, the extremal functions ϕ+

1 , · · · , ϕ+ ℓ ∈ Φ,

the subextremal functions Φ−

S ⊂ Φ.

Example with k = 4 : Θ = ({s1}, {s2}, {s3, s4}) Θ = ({s1, s2}, {s3, s4})

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 Pk be the set of partitions of {s1, · · · , sk}. We note s = (s1, . . . , sk). Theorem For τ = (τ1, . . . , τℓ) ∈ Pk, A ⊂ Cℓ

0 and B ⊂ Mp(C0) measurable

P

• Θ = τ, (ϕ+

1 , . . . , ϕ+ ℓ ) ∈ A, Φ− S ∈ B

• =
• Cℓ

1{∀j∈[

[1,ℓ] ], fj>τj ∨j′=jfj′}1{(f1,··· ,fℓ)∈A}

P[{Φ ∈ B} ∩ {∀φ ∈ Φ, φ <S ∨ℓ

j=1fj}] µ(df1) · · · µ(dfℓ)

Furthermore, the law νs of η(s) is equal to νs =

τ∈Pk ντ s with

ντ

s(dy)

= exp

• − µ(f(s) < y)
• ℓ
• j=1

µ

• f(sτ c

j ) < yτ c j , f(sτj) ∈ dyτj

• .

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 ) = (fj).

Finally, set η(t) = |Θ|

j=1 ϕ+ j (t) φ∈Φ−

s φ(t). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 16 / 39

Conditional distribution of max-i.d. random fields

Main theorem

Theorem Let s ∈ T k and y ∈ (0, +∞)k.

1

For all τ ∈ PK, P[Θ = τ | η(s) = y] = dντ

s

dνs (y).

2

Conditionally on η(s) = y and Θ = τ, the extremal functions (ϕ+

j )1≤j≤|τ| are independent and ϕ+ j has distribution

µ

• df | f(sτj) = yτj, f(sτ c

j ) < yτ c j

• .

3

The conditional distribution of Φ−

s given η(s) = y, Θ = τ and

(ϕ+

j ) = (fj) is a PPP with intensity 1{f(s)<y}µ(df).

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 17 / 39

Conditional distribution of max-i.d. random fields

The regular case

The model is called regular at s if µ(f(s) ∈ dz) = λs(z) dz. The conditional hitting scenario distribution is then P[Θ = τ | η(s) = y] = 1 C(s, y)

|τ|

• j=1
• {uj<yτc

j }

λ((sτj ,sτc

j )(yτj, uj)duj.

If the model is regular at (s, t) P[ϕ+

j (t) ∈ dz | η(s) = y, Θ = τ]

=   1 C′

• u<yτc

j

λ(sτj ,sτc

j ,t)(yτj, u, z)

λs(y) du   dz.

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 18 / 39

Conditional distribution of max-i.d. random fields

Conditional sampling for Brown-Resnick processes

A Brown-Resnick process is a 1-Fréchet max-stable process η(t) =

• i≥1

UieWi(t)−γ(t)/2 with {Ui, i ≥ 1} ∼ PPP(u−2du) and (Wi)i≥1 i.i.d. copies of a Gaussian process with stationary increment and variogram γ. If the covariance of W(t) is nonsingular, η is regular at t : λt(z) = Ct exp

• −1

2 log zTQt log z + LT

t log z

• k
• i=1

z−1

i

. Step 1 : a Gibbs sampler for the conditional hitting scenario. Step 2 : extremal functions are (conditionned) log-normal process.

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 19 / 39

Conditional distribution of max-i.d. random fields

Conditional sampling for Brown-Resnick processes

Conditional sampling for 1D Brown-Resnick process driven by fractional Brownian motion : Numerical study for a 2D Brown-Resnick process calibrated on real data : extreme temperature and rainfall in Switzerland. Mored details in Mathieu’s talk tomorrow !

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 20 / 39

Conditional distribution of max-i.d. random fields

Perspectives

Perfect simulation of the conditional hitting scenario ? Further classes of models with effective conditional simulations ? See Wang & Stoev (spectrally discrete processes) and Oesting (M3 processes). Characterization of the max-stable processes with the Markov property ? Maximum likelihood estimation for multivariate max-stable distributions :

consistency, asymptotic normality, efficiency of the MLE ? a criterion for local asymptotic normality ? design of efficient algorithm ? an EM procedure with the hitting scenario as a hidden variable ?

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 21 / 39

Strong mixing properties of max-i.d. processes

Structure of the talk

1

Overview of the research interests

2

Conditional distribution of max-i.d. random fields

3

Strong mixing properties of max-i.d. processes

4

Intermediate regime for aggregated ON/OFF sources

5

A stochastic gradient algorithm for the p-means

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 22 / 39

Strong mixing properties of max-i.d. processes

Motivations

Statistics of max-stable process based on non i.i.d. observations but rather on stationary weakly dependent observations. Recent results for ergodic and mixing properties of stationary max-stable and max-i.d. processes :

⊲ Weintraub (’91) Sample and ergodic properties of some min-stable processes. ⊲ Stoev (’10) Max-stable processes : representations, ergodic properties and statistical applications. ⊲ Kabluchko & Schlather (’10) Ergodic properties of max-infinitely divisible processes.

Ergodicity and mixing are important to derive strong law of large numbers but not enough to get central limit theorems.

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 23 / 39

Strong mixing properties of max-i.d. processes

Motivations

CLTs for stationary weakly dependent processes are available under strong mixing assumptions. We consider here β-mixing (Volkonskii et Rozanov ’59) : for random variables X1, X2, β(X1, X2) = P(X1,X2) − PX1 ⊗ PX2var. Consider a continuous max-i.d. process η on a locally compact set T such that

• η(t)
• t∈T

L

=

φ∈Φ

φ(t)

• t∈T,

with Φ ∼ PPP(µ). with µ the exponent measure on C0(T) = C(T, [0, +∞)) \ {0}. For disjoint subsets S1, S2 ⊂ T, can we get an estimate for β(S1, S2) = β(η|S1, η|S2) ?

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 24 / 39

Strong mixing properties of max-i.d. processes

A natural decomposition of the point process

For any S ⊂ T, Φ = Φ+

S ∪ Φ− S with

Φ+

S

= {φ ∈ Φ; ∃s ∈ S, φ(s) = η(s)} Φ−

S

= {φ ∈ Φ; ∀s ∈ S, φ(s) < η(s)}.

FIGURE: Realizations of the decomposition Φ = Φ+

S ∪ Φ− S , with S = [0, 5]

(left) or S = {3} (right).

Clearly for s ∈ S, η(s) =

φ∈Φ+

S φ(s) whence

β(S1, S2) ≤ β(Φ+

S1, Φ+ S2).

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 25 / 39

Strong mixing properties of max-i.d. processes

A simple upper bound for β(S1, S2)

Theorem The following upper bound holds true β(S1, S2) ≤ 2 P[Φ+

S1 ∩ Φ+ S2 = ∅] ≤ 2

• C0

P[f <S1 η, f <S2 η] µ(df). In the particular case when η is a 1-Fréchet process, β(S1, S2) ≤ 2 [C(S1) + C(S2)] [θ(S1) + θ(S2) − θ(S1 ∪ S2)] with C(S) = E

• supS η−1

and θ(S) = − log P

• supS η ≤ 1
• the areal

coefficient (Coles & Tawn ’96).

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 26 / 39

Strong mixing properties of max-i.d. processes

A CLT for max-i.d. random fields

Theorem Let η be stationary max-i.d. on Zd and define X(h) = g(η(t1 + h), . . . , η(tp + h)), h ∈ Zd. Assume that there is δ > 0 such that E[X(0)2+δ] < ∞ and β(η(0), η(h)) = o(h−b) for some b > d max

• 2, 2 + δ

δ

• .

Then Sn =

h≤n X(h) satisfies the central limit theorem :

c−1/2

n

• Sn − E[Sn]
• =

⇒ N(0, σ2) with cn = card{h ≤ n} and σ2 =

h∈Zd cov(X(0), X(h)).

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 27 / 39

Strong mixing properties of max-i.d. processes

Perspectives

Systematic study of other notions of dependence. In the spirit of the result by Samorodnitsky, Stoev, Kabluchko : η ergodic iff η generated by a null flow, we conjecture η beta-mixing iff η generated by a dissipative flow ? Relationships with the tesselation C(x) = {y ∈ Rd; Φ+

x ∩ Φ+ y = ∅},

x ∈ Rd. It holds C(0) bounded iff η generated by a dissipative flow, C(0) "with 0 density" iff η generated by a null flow.

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 28 / 39

Intermediate regime for aggregated ON/OFF sources

Structure of the talk

1

Overview of the research interests

2

Conditional distribution of max-i.d. random fields

3

Strong mixing properties of max-i.d. processes

4

Intermediate regime for aggregated ON/OFF sources

5

A stochastic gradient algorithm for the p-means

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 29 / 39

Intermediate regime for aggregated ON/OFF sources

Motivations

Modelisation of packet traffic on high-speed links : traffic measures show LRD and self-similarity. Many models developed to explain these features, considering traffic as the aggregation of a large number of streams generated by independent sources :

⊲ Mikosh & al. (’02), Is network traffic approximated by stable Lévy motion or fractional Brownian motion ? ⊲ Mikosch & Samorodnitsky (’07), Scaling limits for cumulative input processes.

Suitable heavy-tail assumptions and scalings account for long range dependence and self-similarity. Asymptotic theory (when both the number of sources and the time scale go to infinity) is very rich :

Many results in the so-called "heavy traffic" or "light traffic regime". We focus here on the intermediate scaling.

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 30 / 39

Intermediate regime for aggregated ON/OFF sources

The ON/OFF model

A single ON/OFF source

Input process I(t) Workload process W(t) 1 Time t X0 Y0 X1 Y1 X2 Y2 X3 Y3

Bivariate renewal (Xi,Yi)i≥0. Input and workload process :

I(t)=1[0,X0)(t)+

n≥0 1[Tn,Tn+Xn+1)(t),

W(t)= t

0 I(s)ds.

Aggregation of sources Cumulative workload of a network fed by m sources : Wm(t) =

m

• j=1

W j(t). Asymptotic of the form Wm(at) − E[Wm(at)] b(a, m) = ⇒ ??? when both a, m → +∞ ?

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 31 / 39

Intermediate regime for aggregated ON/OFF sources

The ON/OFF model in intermediate scaling

Theorem Assume the following heavy-tail assumptions ¯ Fon ∈ RV(−αon), ¯ Foff ∈ RV(−αoff) with 1 < αon < αoff < 2. In the intermediate scaling, a → +∞, m → +∞, ma¯ Fon(a) → µ cαon−1 (c > 0), the following convergence of processes holds in C(R+, R) : Wm(at) − E[Wm(at)] a = ⇒ σ1(αon)µoff µ c PH(t/c), with PH fractional Poisson motion of Hurst index H = (3 − αon)/2.

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 32 / 39

Intermediate regime for aggregated ON/OFF sources

The fractional Poisson motion PH

Definition of fractional Poisson motion :

PH(t)=

1 σ1(αon)

• R×R+

t

0 1[x,x+u](y) dy (N(dx,du)−dx αonu−αon−1du)

where N(dx, du) ∼ PPP(dx αonu−αon−1du) on R × R+. Properties of PH :

infinitely divisible process with the same covariance structure as FBM with Hurst index H, but positive skewness ; long range dependence ; aggregate self-similarity ; paths are "almost" H-hölder continuous.

PH realizes a Bridge between H-FBM and αon-stable Lévy motion (interpolation between "heavy traffic" and "light traffic" limits).

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 33 / 39

Intermediate regime for aggregated ON/OFF sources

A moment measure approach

Theorem Let (Tn)n≥1 be a stationary renewal sequence with arrival law F, ¯ F ∈ RV(−α), 1 < α < 2. Then, the k-th centered moment measure Mk of the point process {Tn, n ≥ 1} satisfies lim

a→∞

(α − 1)µk+1 ¯ F(a)ak+1

• Mk[f(·/a)] = (−1)kQk[f],

where f is a regular test function on Rk

+ and Qk is the limit measure

Qk(dx) = dx1 · · · dxk | max(xi) − min(xi)|α−1 .

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 34 / 39

Intermediate regime for aggregated ON/OFF sources

Perspectives

Generalize the results to regenerative random measure, i.e. random mesaures satisfying the renewal property ξ(·) L = η0(·) + ξ(· − T1) where ξ and (η0, T1) are independent. Framework covering most usual models : ON/OFF , cluster models, renewal/reward, semi-Markovian models .... Computations of all moment measures via renewal theory and recursive formulas. Asymptotics under heavy tail assumptions ?

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 35 / 39

A stochastic gradient algorithm for the p-means

Structure of the talk

1

Overview of the research interests

2

Conditional distribution of max-i.d. random fields

3

Strong mixing properties of max-i.d. processes

4

Intermediate regime for aggregated ON/OFF sources

5

A stochastic gradient algorithm for the p-means

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 36 / 39

A stochastic gradient algorithm for the p-means

Motivations

Computation of the p-mean (p ≥ 1) of a probability measure µ on a Riemanian manifold (M, ρ) : ep = argminx∈M

• M

ρ(x, y)p µ(dy). Condition for existence and unicity of the p-mean by Afsari ’10 :

(A1) µ has support in a geodesic ball of radius r < r0(p, M), (A2) if p = 1, the support of µ is not contained in a geodesic.

Computation of the p-mean ?

Case p = 2 : Riemanian barycenter or Karcher mean e2. A gradient descent algorithm by Le ’04. Case p = 1 : Fermat-Werber problem for the median e1. Algorithms by Weiszfeld ’37, Fletcher ’09, Yang ’10.

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 37 / 39

A stochastic gradient algorithm for the p-means

A stochastic gradient algorithm (convergence)

Theorem Suppose Afsari conditions (A1)-(A2) are satisfied. Let (Pk)k≥0 be an i.i.d. sequence on M with distribution µ. Let (tk)k≥1 be positive reals such that tk ≤ δ1(p, M, µ),

• k≥1

tk = ∞ and

• k≥1

t2

k < ∞.

Then, the sequence (Xk)k≥0 defined by X0 = P0, Xk+1 = expXk

• − tk+1gradXkρp(Pk+1, ·)
• ,

k ≥ 0. converges almost surely and in L2 to ep. Remark : with M = Rd, p = 2, tk =

1 2k , Xk = k i=1 Pi.

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 38 / 39

A stochastic gradient algorithm for the p-means

A stochastic gradient algorithm (fluctuations)

Theorem Suppose Hp : x →

• ρp(x, y)µ(dy) is C2 on a neighbourhood of ep.

Set tk = δ

k ∧ δ1 with δ > δ2(p, M, µ). Then the normalized process

Yn(t) = [nt] √n exp−1

ep (X[nt]),

t ≥ 0, converges weakly in D((0, +∞), TepM) to the solution of the SDE y0+ = 0, dyt = (yt − δ∇dHp(yt, ·)♯)t−1dt + dBt, t > 0, where ∇dHp(yt, ·)♯ is the dual vector of ∇dHp(yt, ·) and (Bt)t≥0 is a Brownian motion with covariance Γ = E[gradepρp(P1, ·) ⊗ gradepρp(P1, ·)].

C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 39 / 39