MATHF302 Probabilités II. Fiche d`exercices 3.

F X F ←(u) := inf{x:F(x)≥u}
X∼F←(U)U∼U(0,1)
(Ci,j)i,j≥1π
σAn:= σ(Cn,k,k ≥1)
n≥1
(Cn)n≥1π
A⊥⊥ σ(C1,...,Cn)n≥1A⊥⊥ σ(C1,C2, . . .)
•Xn−→ X
• ∀ε > 0, P (∃n0,∀n≥n0,|Xn−X|< ε) = 1
• ∀ε > 0, P (|Xn−X| ≥ ε)=0
• ∀ε > 0,limn→∞ P(maxk≥n|Xk−X| ≥ ε)=0
Xn
P
−→ X(nk)k≥1Xnk−→ X
(Xn)
lim
α→∞ sup
n
E|Xn|I{|Xn|> α}= 0.
(Xn)n≥1Xn−→ X
supnE|Xn|<∞
X
∀α > 0Xα
n→XαZα=ZI{|Z| ≤ α}
EXn→EX
Xn
L1
−→ X
supnE|Xn|1+<∞ > 0Xn
L1
−→ X
1≤p, q ≤ ∞ 1/p + 1/q = 1 X∈LpY∈Lq
E|XY | ≤ (E|X|p)1/p(E|X|q)1/q
(Xn)EXp
n<∞p > 0Xnn−1/p −→ 0
(Xn)∼B1,1/2n
Ln:= 
1Xn= 0, Xn+1 = 1
2Xn= 0, Xn+1 = 0, Xn+2 = 1
P∞
n=1 2−rn<∞=⇒P(Ln≥rn,)=0
P(Ln≥(1 + ε) log2(n) ) = 0 ∀ε > 0
PSε∈Q∩]0,∞[{Ln≥(1 + ε) log2(n),}= 0

lim supn→∞
Ln
log2(n)≤1,
∗
lim sup
n→∞
Ln
log2(n)= 1,
(Xn)n≥1EX1=µVar(X1) = σ2<∞
¯
Xn=1
nPn
i=1 Xi¯
Xn2→µ
ε > 0
Pmax
n2<k≤(n+1)2|¯
Xn2−¯ Xk|> ε = 0.
¯
Xn
p.s.
→µ