Feuille 11 - Loi des grands nombres

(Xn)n∈N
(Ω,F,P)fR R (Xn)n∈N
X(f(Xn))n∈N
f(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|
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

(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