Suites de variables aléatoires. - Université Pierre et Marie CURIE

ARA
M x ∈A x ≤M A
Asup A
Asup A S
x∈A x ≤S
M x ∈A x ≤M S ≤M
A A
sup A= +∞sup ∅=−∞
ARSsup A=S
x∈A x ≤S
ε > 0x∈A S −ε<x
(an)n≥1p≥1sp= sup{an:
n≥p}
(sp)p≥1
[−∞,+∞]
(an)n≥1lim an
(an)n≥1lim an=−∞ lim an= +∞
lim an(an)n≥1
(bk)k≥1(an)n≥1
n1< n2< . . . k ≥1
bk=anklimk→∞ bk≤limn→∞ an
(an)n≥1lim an
(an)n≥1c
lim an> c c0> c an> c0
n
lim an< c c0< c an< c0
n
(Xn)n≥1X= limn→∞ Xn
a, b a < b {ω∈Ω : a <

X < b}
(Xn)n≥11
Plim Xn
log n>1= 0
X1, X2, . . .
Plim Xn
log n<1= 0
lim Xn
log n
lim Xn0
l
l
X
X
(Xn)n≥1
X f :R→R
(f(Xn))n≥1f(X)