Loi Forte des Grands Nombres

(Xn)n≥1L1
Sn=X1+. . . +Xn
Sn
n→0p.s.
(Yn)n≥1
Yn−→ 0p.s. ⇐⇒ Wn= sup
k≥n
|Xk|P −→ 0
Sn
n→0p.s.
∀ε > 0, P µsup
k≥n¯¯¯¯
Sn
n¯¯¯¯> ε¶−→
n→+∞0
'
&
$
%
n≥1ε > 0
εP µsup
k≥n¯¯¯¯
Sn
n¯¯¯¯> ε¶≤°
°
°
°
Sn
n°
°
°
°1
Sn
n
L1
→0Sn
n→0p.s.
(m, n) 1 ≤m≤n
ε > 0
εP µsup
m≤k≤n¯¯¯¯
Sk
n¯¯¯¯> ε¶≤°
°
°
°
Sm
m°
°
°
°1
Tn= sup ½1≤k≤n , ¯¯¯¯
Sk
k¯¯¯¯> ε¾sup ∅=−∞
A=½sup
m≤k≤n¯¯¯¯
Sk
k¯¯¯¯> ε¾
A={Tn≥m}=X
m≤k≤n
{Tn=k}εP (A) = εX
m≤k≤n
P(Tn=k)
Tnk m ≤k≤n
εP (Tn=k)≤Z{Tn=k}¯¯¯¯
Sk
k¯¯¯¯dP =Z{Tn=k , Sk/k>0}
Sk
kdP
| {z }
B
+Z{Tn=k , Sk/k<0} −Sk
kdP
| {z }
C

B C
B=1
k
k
X
j=1 Z{Tn=k , Sk/k>0}
XjdP
Xn
k
Z{Tn=k , Sk/k>0}
X1dP
m m ≤k
B=1
m
m
X
j=1 Z{Tn=k , Sk/k>0}
X1dP =Z{Tn=k , Sk/k>0}
Sm
mdP
C
C=Z{Tn=k , Sk/k<0} −Sm
mdP
εP (Tn=k)≤B+C=Z{Tn=k}¯¯¯¯
Sm
m¯¯¯¯dP
k
εP (A)≤X
m≤k≤nZ{Tn=k}¯¯¯¯
Sm
m¯¯¯¯dP ≤Eµ¯¯¯¯
Sm
m¯¯¯¯¶=°
°
°
°
Sm
m°
°
°
°1
L1