Feuille de TD numéro 1

(Xn, Yn)XnYn
X Xn−Yn0
(Xn, Yn)XnYn
(Xn, Yn)
Xn≥0
E(Xn)
Xn
XnX t ∈ReitXn
eitX (cos(Xn),sin(Xn)) (cos(X),sin(X))
(Xn)E[min(|Xn|,1)] →0
Xnx Xnx
Sn√nSn
Sn0
TnRd√n(Tn−θ)
Tnθ
(Xn) (Yn)
XnX Ync
(Xn, Yn) (Xn, c)
XnYncX
(X1, . . . , Xn)
m σ2>0
XnS2
n
Tn=√n(Xn−m)
Sn
Xn
(n, λ/n)λ
(Xn)Rd(Xn)
N(0,Γ) A d ×d(AXn)
N(0, AΓAT)

PE[|Yn|]<∞Pn
i=1 Yi
(Xn)
(Xn
P
−→ 0) ⇐⇒ (E[min(|Xn|,1)] →0
XRdN(m, R)
u∈RduTXN(uTm, uTRu)
(X1, . . . , Xn) [−θ, θ]θ
E[X2]θ
X2:= 1
nX2
1+··· +X2 n.
Tn= max1≤i≤n|Xi|(|X1|/θ, . . . , |Xn|/θ)
Tn/θ
α > 0ε > 0
nαP(|Tn−θ| ≥ ε/√n)→0, n → ∞.
(X(1), . . . , X(n))U([0,1])
δi=X(i)−X(i−1) ,1≤i≤n,
X(0) = 0
(X(1), . . . , X(n))
f
f(X(1), . . . , X(n)) = X
σ
f(Xσ(1), . . . , Xσ(n))
(δ1, . . . , δn)
δin(1 −u)n−11[0,1](u)i
(δi, δj)i6=j
n(n−1)(1 −u−v)n−21∆2(u, v) ∆kRk
∆k=nx= (xi)∈Rk:xi≥0, x1+··· +xk≤1o.
(Y1, . . . , Yn+1)
Si=Y1+··· +Yiξ= (ξi)
ξi=Si/Sn+1,1≤i≤n.
ξ
λ > 0