Graphes aléatoires

G0

n n

n n
p
Ω
P:P(Ω) →[0,1]
P[Ω] = 1
PhS+∞
n=0 Ani=P+∞ n=0 An(An)n≥0

px=P[{x}]
Ω
X: Ω →RX=a X a ∈R
P[X=a] = PX−1({a})A B
P[A∩B] = P[A]P[B]
(pX(k) = P[X=k])k∈N
X
X
ΩP[X=a]>0ω∈ΩX(ω) = a
X
N
E[X] =
+∞
X
k=0
kP[X=k].
ω∈ΩX(ω)≥E[X]
∀ω∈Ω, X(ω)<E[X]E[X] =
PE[X]−1
k=0 xP[X=k]<E[X]PE[X]−1 k=0 P[X=k]≤E[X]E[X]<E[X]
(an)n∈N
P+∞
n=0 anXn
(pX(k))kX
GX(x) =
+∞
X
k=0
P[X=k]xk
x∈[−1,1]

G(n)
Xn GX
k!pk=G(k)
X(0).
n
n
n µn=E[Xn]
µ3=E[Xn] = "xd
dx(n)
GX(x)#x=1 ,
xd
dx(n)x x
n
µ1=E[X] = G0
X(1)
X Y
GX+Y=GXGY.
GX+Y(z) =
+∞
X
k=0
P[X+Y=k]zk
=
+∞
X
k=0 X
x,y
P[{x+y=k}∩{X=x}∩{Y=y}]zk
=
+∞
X
k=0 X
x,y
P[x+y=k]P[X=x]P[Y=y]zk
=
+∞
X
k=0 X x+y=k
P[X=x]zxP[Y=y]zy
= +∞
X
x=0
P[X=x]zx! +∞
X
y=0
P[Y=y]zy!
=GX(z)GY(z).