6 - CMAP, Polytechnique

γ(dx) = exp(−x2/2) dx/√2π.
Φ(t) = RReitx−x2/2dx/√2π
N(0,1).
r > 0 Γ(r) = R∞
0e−ttr−1dt Γr,λ
γr,λ(t) = λr
Γ(r)e−λttr−11R+(t). r > 0, λ > 0.
Γr,λ.
X Y Γr,λ Γs,λ
X+YΓr+s,λ.
X1, . . . , Xnλ.
X1+. . . +Xn.
X1, . . . , XnN(0,1). X2
1Γ
X2
1+. . . +X2
nχ2
(X1, . . . , Xn)
X= (X1, . . . , Xn).
X
A∈Mn(R)B∈RnAX +B
P∈Mn(R)X P X
X
||X|| S(n−1) Rn
S(n−1)
(X1, X2, X3)
A=
213
156
369
X3−α1X1−α2X2X3
X1X2
1
un=e−n
n
X
k=0
nk
k!.

p
400 40
p5
ˆp p
0,005 95%.
n= 400
g0≤g≤1m=R1
0g(x)dx
X Y [0,1]
U=1Y≤g(X), V =g(X)W=g(X) + g(1 −X)
2.
U, V W
m
gE(g(X)g(1 −X)) ≤m2
(g(x)−g(y))(g(1 −x)−g(1 −y)) ≤0
x, y
(Xi)i≥1[0,1]
An=1
2n
2n
X
i=1
g(Xi)Bn=1
2n
n
X
i=1
(g(Xi) + g(1 −Xi))
m
g(x) = x2
n AnBn
%m%
(Xi:i≥1)
1
n→ ∞
√nPn
i=1(Xi−1)
Pn
i=1 Xi
.
AnBn
AnA Bnb
An+BnAnBnA+b Ab
X a > 0
E[exp iuX] = exp (−a|u|)
(Xn, n ≥1)
a > 0Sn=Pn
k=1 Xk
Sn
√n, n ≥1
Sn
n2, n ≥1
Sn
n, n ≥1

S2n
2n−Sn n.
n(Xn,m)1≤m≤n
P(Xn,m = 1) = pn,m,P(Xn,m = 0) = 1 −pn,m,
lim
n→∞
n
X
m=1
pn,m =λ∈]0,∞[,
lim
n→∞ max
1≤m≤npn,m = 0.
Sn=Pn
m=1 Xn,m
λ
Πn
m=1zm−Πn m=1wm
≤
θn−1Pn
m=1 |zm−wm|ziwi≤θ