Convergence en loi, TCL.

(Xn)n∈N
(Ω,F,P)fR R (Xn)n∈N
X(f(Xn))n∈N
f(X)
f(Xn)
Xnf ϕ
R R ϕ◦f
(Xn)n∈NXlim E[ϕ◦
f(Xn)] = E[ϕ◦f(X)] lim E[ϕ(f(Xn))] = E[ϕ(f(X))]
ϕ(f(Xn))n∈Nf(X)
(Xn)n≥1
a
(Xn)n≥1
Sn=Pn
k=1 Xk
(1
√nSn)n≥1(1
nSn)n≥1(1 n2Sn)n≥1
φ(t) =
e−|t|
ε > 0fεf(x) = ]a−ε,a+ε[c(x) + 1
ε|x−a|]a−ε,a+ε[
(E[fε(Xn)])n≥1E[fε(a)] = 0
P(|Xn−a≥ε|) = E[]a−ε,a+ε[c(Xn)] ≤E[fε(Xn)],
(Xn)n≥1a
1
√nSn
φ(t) = E[e
it
√nSn] = E[e
it
√nX1]n=e−|t|√n,

Xne−|t|√nn≥11{0}(t)
0
(1
√nSn)n≥1
1
nSnφ(t) = e−|t|(1 nSn)n≥1
(Sn
n−S2n 2n)n≥1
0Sn
n−S2n 2n
φ(t) = E[eit
2n(X1+···+Xn−(Xn+1+···+X2n))] = e−|t|1
0
1
n2Snφ(t) = e−|t|
n1
(1
n2Sn)n≥10
X(Xn)n∈N(Yn)n∈N
t∈Ra > 0n∈N
|φXn+Yn(t)−φXn(t)| ≤ 2P(|Yn> a|) + E[]−∞,a](|Yn|)|eitYn−1|].
(Xn)n∈NX(Yn)n∈N
(Xn+Yn)n∈NX
(Xn)n∈NX
(Xn−X)n∈N
t∈Ra > 0n∈N
|φXn+Yn(t)−φXn(t)|=|E[eitXn(eitYn−1)]| ≤ E[|eitYn−1|]
=Z{|Yn|>a}|eitYn−1|dP+Z{|Yn|≤a}|eitYn−1|dP
≤2P(|Yn|> a) + Z{|Yn|≤a}|eitYn−1|dP
Ynε > 0a0>0
y≤a0|eit −1| ≤ ε n0∈Nn≥n0
P(|Yn|> a0)≤ε|φXn+Yn(t)−φXn(t)| ≤ 3ε(Xn)n∈N
X n1n0
n≥n1|φXn(t)−φX(t)| ≤ ε n ≥n1
|φXn+Yn(t)−φX(t)| ≤ 4ε
X X −X Xn=−X
XnX Xn−X−2X

(Xn)n≥1
Sn=Pn
k=1 XkSn
e−nPn
k=0
nk
k!n≥1
X
E[|X−inf(X, a)|]≤E[X{X≥a}]≤(E[X2]P(X≥a))1/2.
(Xn)n≥1
Sn=Pn
k=1 XkYn=Sn−n
√n
n≥1E[Y2
n]P(Y− n≥a)≤1
a2.
(Yn)n≥1
(Y−
n)n≥1Y−(inf(Y− n, a))n≥1
inf(Y−, a)
(E[Y−
n])n≥1E[Y−]
E[Y−]E[Y−
n]n≥1
lim
n→+∞
√2πnnne−n
n!= 1.
(pn)n≥0]0,1[ limn→+∞npn=λ > 0
(Xn)n≥0n Xn∼ B(n, pn)X
λ(Xn)n≥0X
(Xn)n≥0N(0,1)
Yn=1
nPn
k=1 √kXk
(Xn)n≥0X
φXn(t)−−−−→
n→+∞φX(t)t∈RφX
X

φXn(t) = 1−pn+pneitn
= exp nln(1 −pn+pneit)
= exp −npn(1 −eit) + npnnn−−−−→
n→+∞0
∼nexp −npn(1 −eit)
−−−−→
n→+∞eλ(eit−1) =φX(t).
Yn√kXk
Xn
t∈R
φYn(t) =
n
Y
k=1
φXk √k
nt!=
n
Y
k=1
e−kt2/n2=e−n(n+1)
2n2t2−−−−→
n→+∞e−t2/2.
(Yn)n≥1
(Xn)n≥0
E[X1] = 30 V ar(X1) = 900 Xn
(n−1) n−Sn=
X1+. . . XnSnn−
P[S50 ≤24 ×60].
P[S50 ≤24 ×60] = PS50 −50 ×30
30 ×√50 ≤24 ×50 −50 ×30 30 ×√50 ≈F−−2
√50,
F
F−−2
√50 ≈F(−0.283) = 1−F(0.283)
F(0.283) ≈0.61 P[S50 ≤24 ×60] ≈0.39