Suites de variables aléatoires.

(Ω,F,P)
(An)n≥1
(An)n≥10
(An)n≥1L20
(An)n≥10
An0
P(An>1
2) = P(An) 0 P(An)
0ε > 0P(An> ε)≤P(An) 0
limn→∞ P(An)=0
E[An] = P(An) limn→∞ P(An) = 0
ω∈Ω ( An(ω))n≥10
0ω
lim inf Ac
nlim sup An
P(lim sup An)=0
(Xn)n≥1Xn
X Y X =Y
X
Y Lpp∈ {1,2}
(Xn)n≥1X Y
ε > 0P(|X−Y|> ε)=0
n≥1
P(|X−Y|> ε)≤P|X−Xn|>ε
2|Xn−Y|>ε 2
≤P|X−Xn|>ε
2+P|Xn−Y|>ε 2.
n+∞0 (Xn)n≥1
X Y P(|X−Y|> ε)=0
P(X6=Y) = P [
k≥1
|X−Y|>1
k!= lim
k→∞
P|X−Y|>1
k= 0.

X Y
(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
k≥1k−1
n1<· · · < nk−1P|Xn−X|>1
k→0n→ ∞
n=nk> nk−1
P|Xnk−X|>1
k≤1 2k.
lim
K→∞
K
X
k=1
P|Xnk−X|>1
k=X
k≥1
P|Xnk−X|>1
k<+∞.
X
k≥1
P|Xnk−X|>1
k= lim
K→∞
K
X
k=1
P|Xnk−X|>1
k= lim
K→∞
E"K
X
k=1
{|Xnk−X|>1
k}#
=E"lim
K→∞
K
X
k=1
{|Xnk−X|>1
k}#=E"X
k≥1
{|Xnk−X|>1
k}#.
Pk≥1{|Xnk−X|>1
k}
ω k |Xnk−X|>1
k
k|Xnk−X| ≤ 1
k|Xnk−X| → 0k→ ∞

(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}
[
n≥k
Ank
k
P [
n≥k
An!&P \
k≥1[ n≥k
An!=P(lim sup An).
P [
n≥k
An!≤X
n≥k
P(An).
P(An)
0k
P(lim sup An)
X(Ω,F,P)
X
n≥0
nP(n≤X < n + 1) <+∞ ⇔ E[X]<+∞.
X
n≥1
P(X≥n)<+∞ ⇔ E[X]<+∞.
n≥0
nn≤X<n+1 ≤Xn≤X<n+1 ≤(n+ 1) n≤X<n+1.
nP(n≤X < n + 1) ≤E[Xn≤X<n+1]≤(n+ 1)P(n≤X < n + 1).

n≥0
X
n≥1
nP(n≤X < n + 1) ≤E[X]≤1 + X
n≥0
P(n≤X < n + 1).
(ak,n)k≥0,n≥0:= P(n≤X < n+1) k<n
k
X
n≥0X
k≥0
ak,n =X
n≥0
P(n≤X < n + 1)sumk≥0k<n =X
n≥0
nP(n≤X < n + 1).
n k
X
k≥0X
n≥0
ak,n =X
k≥0X n>k
P(n≤X < n + 1) = X
k≥0
P(X≥k+ 1) = X
k≥1
P(X≥k).
(Xn)n≥1
Xn
n
P
−→
n→∞ 0
E[|X1|]<+∞Xn
n
L1
−→
n→∞ 0
E[|X1|]<+∞Xn
n
p.s.
−→
n→∞ 0
Xnε > 0
n≥1
P|Xn|
n> ε=P|X1| n> ε=P(|X1|> nε).
n→ ∞
lim
n→∞
P(|X1|> nε) = P(\
n≥1
{|X1|> nε}) = P(|X1|= +∞)=0.
n→+∞E[|X1|]<+∞
E|Xn|
n=E|X1| n=1 nE[|Xn|]→0.
XnXn
n
L1
−→
n→∞ 0
E[|X1|]<+∞

E[|X1|]<+∞ε > 0
X=|X1|/ε P(|X1| ≥ εn)P(|Xn| ≥ εn)
P(lim sup{|Xn| ≥ εn})=0.
Aε
lim sup{|X1| ≥ εn}
∃N≥0,∀n≥N, |Xn|< εn.
Aεlim sup |Xn|/n ≤ε ε = 1/k k ≥1
∩n≥1A1/k
A1/k
lim sup |Xn|/n = 0
X
Xn:= X n ≥1
(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.
E[(Xn−a)2] = E[((Xn−E[Xn]) + (E[Xn]−a))2]
= Var(Xn)+2E[(Xn−E[Xn])(E[Xn]−a)] + (E[Xn]−a))2.
(E[Xn]−a)
E[(Xn−a)2]→0n→ ∞
0≤E[(Xn−a)] ≤pE[(Xn−a)2]→0.
Var(Xn) = E[(Xn−a)2]−(E[Xn]−a)2→0n→ ∞
lim
n→∞
E[Xn] = alim
n→∞ Var(Xn)=0.