Un parcours explicite en théorie multiplicative
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2003
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- mémoire
MÉMOIRE Un parcours explicite en théorie multiplicative présenté pour l'obtention de l'Habilitation à Diriger des Recherches Université de Lille 1 Olivier Ramaré présenté et soutenu publiquement le 6 Juin 2003 devant le jury composé de Jean-Marc Deshouillers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Président Michel Balazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rapporteur Hédi Daboussi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rapporteur Andrew Granville . . . . . . . . . . . . . . . . . . . . . . . . . .
- catégorie des problèmes polynômiaux
- crible de dimension
- constante de ?nirel'man
- parcours explicite en théorie
- tamment sur le problème du zéro de siegel
- cribles
- contribution des bords de l'intervalle
- parcours explicite en théorie multiplicative
20Habilitationvierunivjane2010rsitaire22.3.2T.able.des.mati?res.1.A.p.erturbationtedexpansionehametho.dP7.1.0.1.I.n.tro.duction....3.1.1...........b...ork.Random.30.....lab.........3.eakly.....results...24.........267.1.0.2.P.erturbation.Expansionthe:.Outlines.of.the.Metho.d..........path.....18.maximal..8.1.0.3generalSome.denitions....23.sums.random.ti.................sc...................tro.........metho9.1.0.4.Main.resultscomp.....A.................tro...................eigh.the...........W.to.el10.1.0.5.A.w.ork2.3.4ed-out.example............Limit.Asymptotic.fo.some.endan.ariables.tro.................3.1.2........11.2.Some.limit.resultsDononsrandom.trees.13.2.1.In.tr.o.duction25.a...................I.................3.2.2.di...............3.2.3.o.urn.......813ject2.2.Distances.in.T.rees......30.tation...............I.................3...................17.W13ted2.2.1toIminimalneltro.duction................2.3.3.eigh.path.the.lab...................19.Some.questions..............13.2.2.2.D.i.gital.trees21.Some.Theorems.3.1.b.vior.r.of.w.dep.t.v.23.in.duc.on...............................2313Main2.2.3.Notation.and.metho.dolo.gy......................3.1.3.i.ussi....................15.2.2.4.D.i.stanc.es.in3.2DSToly.urns.....................................3.2.1.n.duction............15.2.2.5.T.ries............26.Em.ed.ng.d...........................27.Asymptotic.osition.f.discrete..............1622.33.2.4Binaryprotreew...............................3.3.fragmen...............................3.3.1.n.duction17.2.3.1.I.n.tro.duction....................30...3.3.2ofHomogeneous.random.fragmen.tation.pro3.4.4cess39.......................Some..31.3.3.3.Exp3.4.3onenmotia.lorksfragmen.tation.probabilit.yduction...............36................32enerating3.3.4generalizedA.pro.ject.wfutureork..............tro.................................3.4.2.examples........36.3.4.A.pattern.matc.hing.probl.e.m....37.G.function.the.Moran.del.........37.Some.w........................36.3.4.1.I4nη = (I−μF )η +ξ ,n+1 n n n+1
ξ :n
μ :
F :n
(I−μF )n
η r rn
F , μn
(X )i,n 0≤i≤n
nX
lim lim sup n cov(X ,X ) = 0;0,n r,n
N→N0 n→+∞
r=N
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N N lim n cov(X ,X ) = 00 n→+∞ 0,n r,nr=N
˜(X ) S :=i,n 0≤i≤n nP Pn n˜ ˜X S := Xi,n n i,ni=1 i=1
1, 2,···n
n m n m
n m
x (U,1−U)
U [0,1]
−xn = 0 1−e x Ux (1−U)x
−xe n l
−l1−e
(U,1−U)
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d
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{y} {v} {φ}t t≥0 t t≥0 t t≥0
{θ} dt t≥0
θ =θ +wt+1 t t+1
wt+1
ˆθt
Tˆ ˆ ˆθ =θ +μL (y −φ θ ).t+1 t t t tt
μ Lt
pursued.iseof,scalartracobtainoftthatvisalgorithmstroleenconoundsautomaticbandtocessing,cproasignalosttication,Resultsidenpapsystemcinthatareerturbationthehissuewhere-dimensionalstep-sizestohcnhasticaregressoronandethemainunknothewnhatime-v43,aryingtparame-toter.andThiskingmoedeltencompassesgettingmancesses,ythedierenparametertisapplications,theincludingacwhihannelbeinqualbization,ttimeTheredelaastyanalysisestimationtandInectributions,hoiscancellationounds[87].kingInthattheesequelinitInisaassumedhthatgoaltheeparametertvonlyariationtheobTeysose,tuseortanhnique,impasAnconsistingductionximationstronestedInm1.0.1structuredkingmethothe(1.2)(1.3)whereationexpansionreferrederturbationaspadaptationandandnoisisreferandomrredector,toctcanheelag-noise.hosenTaoumtracerkdierenthewvys.ariationsisofvtheliteratureparameter,theitofisofcustomaryypto(1.3).usemaconrecursivtheegoalalgorithmtoforbupndatingtracanerrors.estimateinAdirections1vChapitrebThisobtaineddecomp[31,enables42].computationthisexpliciterfordierenmomenapproacandisrelatedOurtities.isMostobtainofxplithesesialgorithmsexpressioncannotbbeforputtracinerror.theoformpurpandwlinearwillheatecregressionreferredmoode"pelyexpansion",linariationsapproectiv(1.3)inyaprorespwithproucaresimpler(1.1)thanobservoriginalerrorwherecess.vparticularofositionthetheparameterof(seeexpressions[31,the87]tsandotherthequanreferences7therein).T˜ ˜θ = (I−μLφ )θ +μLv −w ,t+1 t t t t t+1t
˜ ˆ ˜θ = θ −θ θt t t t+1
u v w˜ ˜ ˜ ˜θ = θ +μθ +θ ,t t t t
u T u u˜ ˜ ˜ ˜θ = (I−μLφ )θ , θ =θ =−θ ,t 0 0t+1 t t 0
v T v v˜ ˜ ˜θ = (I−μLφ )θ +Lv, θ = 0,t t tt+1 t t 0
w T w w˜ ˜ ˜θ = (I−μLφ )θ −w , θ = 0.t t+1t+1 t t 0
u˜{θ }t
v˜{θ }t
w˜{v} {θ } {w}t tt+1
v w˜ ˜θ θt
δ = (I−μF )δ +ξ, δ = 0t+1 t t t 0
tX
= Φ(t,s)ξs
s=0
{F} {ξ} (d×1)t t≥0 t t≥0
Φ(t,s)
(I−μF )(I−μF )···(I−μF ), t>s t t−1 s+1
I, t =s
0,
δ μt
TF =Lφt t t
ξ =Lv ξ =−w .t t t t t+1
¯ ¯F =E(F ) Z =F −F (I−μF )t t t t t t
¯I−μF = (I−μF )+μZ.t t t
(0) (0) (0)¯J = (I−μF )J +ξ, J = 0t tt+1 t 0
(0) (0) (0) (0)¯H = (I−μF )H +μZJ , H = 0t tt+1 t t 0
(0) (0)
δ = J +H .t t t
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Jt
tX
(0)
J = ψ(t,s)ξst+1
s=0
¯ ¯ ¯(I−μF )(I−μF )···(I−μF ), t>s t t s+1
ψ(t,s) = I, t =s
0,
δt
(0) (1) (n) (n)
δ =J +J +···+J +H ,t t t t t
(r) (n)
J , 0≤r≤n Ht t
(0) (0) (0)¯J = (I−μF )J +ξ ; J = 0t tt+1 t 0
(r) (r) (r−1) (r)¯J = (I−μF )J +μZJ ; J = 0, 0≤t<rt tt+1 t t t
(n) (n) (n) (n)
H = (I−μF )H +μZJ ; J = 0, 0≤t<nt tt+1 t t t
q≥ 1 X ={X } (l×1)n n≥0
δ = (δ(r)) Xr∈N
(δ,q) C ={C ,···,C }1 q
1≤m<s≤q m t ,···,t (s−m) t ,···,t t ≤···≤1 m m+1 s 1
t <t +r≤t ≤···≤tm m m+1 s
sup |Cov(X ···X ,X ···X )|≤C δ(r)t ,i t ,i t ,i t ,i s1 1 m m m+1 m+1 s s
i ,···,i1 s
X i Xn,i n
p ≥ 1 n ∈ N G = {G} (d×d)t t≥0
(δ,p(n+2))
X
p(n+2)/2−1(r +1) δ(r)<∞.
r
D (G) j ∈ {1,···,n} 0 ≤p,n
s≤t<∞
X j/2G ···G ≤D (G)(t−s) . i i p,n1 j
pn/j
s≤i <···<i ≤t1 j
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