Variables aléatoires

(Xn)n∈N
E N
(Ω,T)Y
∀ω∈Ω, Y (ω) = XN(ω)(ω)
Y
T
∀n∈N,P(T > n)>0
T(θn)n∈N
θn= P(T=n|T≥n)
T
θn
∀n∈N, θn∈[0 ; 1[
(θn)n∈NP(T≥n)
Pθn
(θn)n∈N
∀n∈N, θn∈[0 ; 1[ Xθn
(θn)n∈N
T

Xn(Ω)
Y(Ω) ⊂[
n∈N
Xn(Ω)
Y(Ω)
y∈Y(Ω)
Y−1({y}) = [
n∈N
{ω∈Ω|N(ω) = n Xn(ω) = y}
Y−1({y}) = [
n∈N
{N(ω) = n}∩{Xn(ω) = y}
T
θnθn∈[0 ; 1]
θn= 1 P(T=n) = P(T≥n) P(T > n)=0
P(T=n) = θnP(T≥n) P(T=n) + P(T≥n+ 1) = P(T≥n)
P(T≥n+ 1) = (1 −θn)P(T≥n)
P(T≥0) = 1
P(T≥n) =
n−1
Y
k=0
(1 −θk)
P(T≥n)−−−−−→
n→+∞0
n−1
X
k=0
ln (1 −θk) = ln n−1
Y
k=0
(1 −θk)!−−−−−→
n→+∞−∞
Pln(1 −θn)
(θn)n∈NPθn
(θn)n∈Nln (1 −θn)∼
n→+∞−θn
Pθn
T
P(T=n) = P(T≥n)−P(T≥n+ 1) = θn
n−1
Y
k=0
(1 −θk)
T
∀n∈N, un=θn
n−1
Y
k=0
(1 −θk)
∀n∈N, un≥0
(un)n∈N
Pn=Qn−1
k=0 (1 −θk)
ln Pn=
n−1
X
k=0
ln(1 −θk)−−−−−→
n→+∞−∞
ln(1 −θk)
(θn)
P0= 1 Pn−Pn+1 =un
n
X
k=0
uk=P0−Pn+1 −−−−−→
n→+∞1
T
P(T=n) = un=θn
n−1
Y
k=0
(1 −θk)
P(T≥n) =
+∞
X
k=n
uk=Pn=
n−1
Y
k=0
(1 −θk)>0
P(T=n|T≥n) = θn
T