BANQUE COMMUNE D`ÉPREUVES

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N C1CN
N
SNN
(SN) = N!
(Ω,A,P) Ω = SNA=P(SN)
SNPΩX E(X)V(X)
X
σ SN
σ1/N!
◦
◦ ◦ N

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i[[1, N]] Cii
C1CN
k[[1, N]] k
k(k+ 1)
N
{1,···, N}
n n = 0
T1
CNN
N−1
T2CNN−2
i[[1, N −1]] TiCN
N−i
∆1=T1∀i∈[[2, N −1]] ∆i=Ti−Ti−1
T=TN−1+ 1
(∆i)i∈[[1,N−1]]
N= 4
n
k
C1C2C3C2C2C1C4C2
C2C3C2C1C1C4C2C4
C3C1C1C4C4C2C3C3
C4C4C4C3C3C3C1C1
T1(ω)=3 T2(ω) = 5 T3(ω) = 6 T(ω)=7.
∀i∈[[2, N −1]] Ti= ∆1+ ∆2+· · · + ∆i
∆i
∆1
n∈NP(∆1> n) ∆1
i∈[[2, N −1]] ∆i
n∈NP(∆i> n) = N−i
Nn∆i
E(∆i) = N
iV(∆i) = NN−i i2
T2n>2
P(T2=n) =
n−1
P
k=1
P(∆2=n−k)P(∆1=k)
n−1
P
k=1 1−1/N
1−2/N k
=N1−1
N"1−1/N 1−2/N n−1
−1#

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P(T2=n) = 2
Nh1−1 Nn−1−1−2 Nn−1i
T2CNN−2
T1T2
T2
T1N−1T2N
T2N−1T1N
T3CNN−3T1T2
T3i∈ {1,2,3}ai
Ti
(a1, a2, a3)
T3
(a1, a2, a3)=(N−2, N −1, N)
(a1, a2, a3)=(N−2, N, N −1)
T
T
CN
(Hn)n>1(un)n∈N∗
∀n>1Hn=
n
X
k=1
1
kun=Hn−ln (n)
T
E(T) = N HNV(T) = N2N
P
k=1
1
k2−N HN
(un)
k>11
k+ 1 6Zk+1
k
dt
t61 k
(un)
∀n∈N∗ln (n+ 1) 6Hn6ln (n)+1
(un)γ
[0,1]
E(T) ˜
N→+∞Nln(N)E(T) = Nln(N) + Nγ +o(N).
V(T)
N2N∈N∗
V(T)
N
α
V(T) ˜
N→+∞αN2V(T)6αN2
X
∀ε > 0P(|X−E(X)|>ε)6V(X)
ε2
N c

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∀ω∈Ω,|T(ω)−Nln (N)|6|T(ω)−E(T)|+N
(|T−Nln (N)|>cN) (|T−E(T)|>N(c−1))
P(|T−Nln (N)|>cN)6α
(c−1)2
α
Nlim
c→+∞
P(|T−Nln (N)|>cN)
ε > 0
lim
N→+∞
P(|T−Nln (N)|>εN ln (N)) = 0
T N ln(N)
32 ln(32) '110
52 ln(52) '205
N= 32
i Cii
i
C10 i
[[0,31]]
>
<>

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πSNP(SN) [0,1]
∀A⊂ SNπ(A) = card (A)
N!∀σ∈ SNπ({σ}) = 1 N!
µnSN
σSNµ({σ})n
σ
ASNµn(A) = P
σ∈A
µn(σ)
n µnπ
d
d(µn, π) = max {|µn(A)−π(A)|, A ⊂ SN}
ASNn∈N∗Enn
A
P(T6n)(En) = π(A)
P(En∩(T6n)) = π(A)P(T6n)
P(En∩(T > n)) 6P(T > n)
µn(A)6π(A) + P(T > n)
SNn∈N∗A A
µnA−πAµn(A)−π(A)
|µn(A)−π(A)|6P(T > n)
∀n∈N∗,06d(µn, π)6P(T > n) lim
n→+∞d(µn, π)
(T > n)
N
N
k∈[[1, n]] Sk
k−1k
S=S1+S2+· · · +SN
N
N N j ∈[[1, N]] Bm
j
m j
(Sk)k∈[[1,N]]
S1
k∈[[2, N]] Sk