Feuille 8 - Suites de variables aléatoires

(Ω,F,P)
(An)n≥1
(An)n≥10
(An)n≥1L20
(An)n≥10
(Xn)n≥1Xn
X Y X =Y
X
Y Lpp∈ {1,2}
(Ω,F,P)X, X1, X2, . . . : (Ω,F,P)→R
(Xn)n≥1
X
1≤n1< n2< . . .
k≥1
P|Xnk−X|>1
k≤1 2k.
k≥1Yk=Xnk(Yk)k≥1
(Xn)n≥1(Yk)k≥1X
(Ω,F,P) (An)n≥1
X
n≥1
P(An)<+∞.
P(lim sup An)=0
lim sup An:= Tk≥1Sn≥kAn={ω∈Ω : {n:ω∈An}estinfini}
X(Ω,F,P)

X
n≥0
nP(n≤X < n + 1) <+∞ ⇔ E[X]<+∞.
X
n≥1
P(X≥n)<+∞ ⇔ E[X]<+∞.
(Xn)n≥1
Xn
n
P
−→
n→∞ 0
E[|X1|]<+∞Xn
n
L1
−→
n→∞ 0
E[|X1|]<+∞Xn
n
p.s.
−→
n→∞ 0
(Xn)n≥1
n≥1a∈R
E[(Xn−a)2] = (E[Xn]−a)2+ Var(Xn).
(Xn)n≥1
a
lim
n→∞
E[Xn] = alim
n→∞ Var(Xn)=0.
(Xn)n≥1(Xn)n≥1
L2X(X2
n)n≥1L1
X2
(Ω,F,P)
(An)n≥1
X
n≥1
P(An) = +∞.
P(lim sup An)=1
x1 + x≤ex
n, m 1≤m≤n
P n
\
k=m
Ac
k!≤exp −
n
X
k=m
P(Ak)!.

m≥1P ∞
\
k=m
Ac
k!= 0
(Xn)n≥1
p∈]0,1[ 1 (Xn)n≥1
1 0
(Xn)n≥1
E[|X1|]=+∞
X1+...+Xn
nn≥1
(xn)n≥1x1+...+xn
nn≥1
lim
n→∞
xn
n= 0.
X
n≥1
P(|Xn| ≥ n)=+∞