Loi des grands nombres.

(Un)n≥1
[0,1] f: [0,1] →R
f(U1) + . . . +f(Un)
n
n+∞f[0,1]
(f(Un))n≥1
1
f2
f(U1)+...+f(Un)
n
n
E[f(U1)] = Z1
0
f(t)dt.
f: [0,1] →Rn≥0
bn: [0,1] →R
∀x∈[0,1], bn(x) =
n
X
k=0 n
kfk nxk(1 −x)n−k.
(bn)n≥0f
∀x∈[0,1],lim
n→∞ bn(x) = f(x).
x∈[0,1] (bn(x))n≥0f(x)
f
K > 0x, y ∈[0,1] |f(x)−f(y)| ≤ K|x−y|
(bn)n≥0f
lim
n→∞ sup{|bn(x)−f(x)|:x∈[0,1]}= 0.

0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.5
1.0
1.5
2.0
2.5
x∈[0,1] bn(x) = E[f(B/n)] B
n x (Xn)n≥1
x
bn(x) = EfX1+. . . +Xn
n.
X1+...+Xn
nn
E[X1] = x f f X1+...+Xn
n
f(x)f[0,1]
fX1+...+Xn
nf(x)f
bn(x) = EfX1+. . . +Xn
n −→
n→∞ f(x).
|bn(x)−f(x)|x
f K
|bn(x)−f(x)|= EfX1+. . . +Xn
n−f(x)
= EfX1+. . . +Xn
n−f(x)
≤KE
X1+. . . +Xn
n−x.
YE[Y]≤

E[Y2]1
2
|bn(x)−f(x)| ≤ KE"X1+. . . +Xn
n−x2#
1
2
=KVar X1+. . . +Xn
n
1
2
=K
√npVar(X1)
=K
√npx(1 −x)
≤K
2√n,
x(1 −x)≤1
4x
kbn−fk∞≤K
2√n.
(bn)n≥1f
pn,k(x) = n
kxk(1−x)n−kn= 10
k∈ {0,...,10}
[0,1] 1
f
f
f b1, b10, b50
b200
π
n N
π'2n
N.

l
θ0π
π/4
θ= 0 θ[0, π[
d
0l d [0, l[
d
[0, π[
[0, l[
Ω = [0, π[×[0, l[
P(dθ dr) = 1
lπ dθdr
lsin θ≥r
P({(θ, r)∈[0, π[×[0, l[: lsin θ≥r}) = 1
lπ Z[0,π[×[0,l[
r≤lsin θdθdr
=1
lπ Zπ
0
lsin θ dθ
=2
π.
l
n n
k∈ {1, . . . , n}
Akk
1
n
n
X
k=1
Ak.

(Ak)k≥1
nE[A1] = P(A1) = 2
π
nN
n'2 ππ'2n N
(Xn)n≥1
{0,1,...,9}
Pn≥1Xn10−n
p≥1l≥0Yl(Xlp+1, . . . , Xlp+p)
(a1, . . . , ap)p{0,...,9}1
1
nCard {l≤n:Yl= (a1, . . . , ap)} −→
n→∞
1
10p.
p≥1r∈ {1, . . . , p}
l≥1Zl= (Xlp+r, . . . , Xlp+r+p−1)
1
nCard {l≤n:Zl= (a1, . . . , ap)} −→
n→∞
1
10p.
(a1, . . . , ap)∈ {0,...,9}p
1
nCard {k≤n:Xk+1 =a1, . . . , Xk+p=ap} −→
n→∞
1
10p.
N⊂[0,1[
x∈[0,1[\N x = 0, x1x2. . . x
∀p≥1,∀a1, . . . , ap∈ {0,...,9},1
nCard {k≤n:xk+1 =a1, . . . , xk+p=ap} −→
n→∞
1
10p.
x
|Xn| ≤ 9n≥1
X
n≥1|Xn10−n| ≤ X n≥1
9.10−n= 1.
Pn≥1Xn10−n
X
0≤X≤1n≥1
k∈ {0,...,10n−1}
Pk
10n≤X < k+ 1 10n=P(X1=kn−1, . . . , Xn=k0) = 10−n,