Indépendance d`événements et de variables aléatoires

(Ω,F,P)
A B
P(A∩B) = P(A)P(B).
(Ai)i∈IF
∀J⊂I, J , P\
i∈J
Ai=Y
i∈J
P(Ai).
A, B P(B)>0
PB(A) := P(A|B) = P(A∩B)
P(B).
A B
PB:F → [0,1]
A7→ PB(A)
A B P(B)>0
A B ⇐⇒ PB(A) = P(A).
Ω = [0,1] P=λ
An:= [
16l62n−12(k−1)
2n,2k−1 2n
n∈N
A B A BcAc
B AcBc
(Ai)i∈I
(Bi)i∈I
Bi=AiBi=Ac
i
(Bi)i∈I
n
ϕ(n) = nY
p|n,p∈P 1−1
p
P
(An)n
PnP(An)<∞P(lim supnAn) = 0
PnP(An) = ∞(An)nP(lim supnAn) = 1

{lim sup An}={Ani.s.} ♦
P( ) = p∈]0,1[
An:= {n}lim sup
n
An:= { }.
P( ) = 1.
Z{Zi}iZ0= 0 i > 0
(Zi+1 =Zi+ 1 p
Zi+1 =Zi−1q= 1 −p
p6=1
2An:= {2n}
P(An) = 2n
npn(1 −p)n∼(4p(1 −p))n
√πn 6am
C
a < 1 0
{Zi}i
(Ai)i∈IP
∀J⊂I, J , ∀Ai∈ Ai,P\
j∈J
Aj=Y
j∈J
P(Aj).
(Xi)i∈I
Xi: (Ω,F,P)→(Ei,Ei)
(Xi)i∈I(σ(Xi))i∈I
X, Y
X Y
∀A∈ E,∀B∈ F,P(X∈A, Y ∈B) = P(X∈A)P(Y∈B)
C D E F
P(X∈A, Y ∈B) = P(X∈A)P(Y∈B)∀(A, B)∈ C × D.
∀f: (E, E)→(R,B(R)) ∀g: (F, F)→(R,B(R))
E(f(X)g(Y)) = E(f(X))E(g(Y)).
∀f: (E, E)→(R,B(R)) ∀g: (F, F)→(R,B(R)) f(X)g(Y)
Rd

(X1, X2, . . . , Xd)P(X1,...,Xd)
(Rd,B(Rd)) PX1⊗ ··· ⊗ PXd
(X1, X2, . . . , Xd)Rr1, . . . , Rrd
∀(t1, . . . , td)∈Rr1× ··· × Rrd, ϕ(X1,...,Xd)(t1, . . . , td) = ϕX1(t1)···ϕXd(td)
r1=··· =rd= 1
F(X1,...,Xd)(t1, . . . , td) = FX1(t1)···FXd(td)
ϕXiFXiXi
U V
[0,1] X=√−2 log Vcos(2πU)Y=√−2 log Vsin(2πU)
{Xn}nF
RnXnRn:= Pn
j=1 {Xj>Xn}{Rn}n
P(Rn=k) = 1
n∀k∈ {1,2, . . . , n}.
(Ω,F,P){Ti}i∈N
Cn=σ[
p>n
TpC∞=\
n∈NCn.
C∞
{Xi}i∈N
Xi: (Ω,F)→(Ei,Ei)
Ti=σ(Xi) := {X−1
i(A) : A∈ Ei} F
{Xn}n
{Xn}n
{Xn}nn→ ∞
A:= {ω∈Ω : limnXn(ω)} ∈ C∞
♦
{Ti}i∈NC∞
A∈ C∞
P(A)∈ {0,1}.
C∞
A:= {Ani.s.}∈A∞An=σ(An) = {∅, An, Ac
n,Ω} {An}n
P(A)∈ {0,1}
{An}nP(lim sup An) = 0 PnP(An)<∞

X, Y
PX+Y=PX? P YϕX+Y=ϕX·ϕY
{Xn}nN(m, σ2)X∼ N(m, σ2/n)S2
n∼χ2(n)X⊥S2 n
X Y L2X+Y⊥X−Y X Y
{Xn}nSn:= Pn
i=1 Xi
X1
Sn
n
P
−−−−→
n→∞
E(X).
X
Sn
n
p.s.
−−−−→
n→∞
E(X).
[0,1] 0 1
f:Rd→Rd>1
I=Z1
0···Z1 0
f(t1, . . . , tn)d(t1, . . . , tn).
X= (X1,··· , Xn) [0,1]dI=E[f(X1,··· , Xn)]
1
N
N
X
i=1
f(X(ωi)) p.s. −−−−→
N→∞ I.
X2m:= E(X)
σ2= Var(X)
Sn−nm
σ√n
L
−−−−→
n→∞ N(0,1).
limn→∞ e−nPn
k=0
nk
k!=1 2
P
{Zn:n∈N}N(pij )
pij =P(Zn+1 =j|Zn=i) = p∗i
j, i >1
δ0j, i = 0 j>0
p∗i
j:= P(Pi
k=0 Yk=j){Yk}kP