X1,X2,...,Xn ˆ Fn : R → [0,1] x → 1 n card{k ≤ n, X k ≤ x} f − g Lp = |f

��
��
��
��
��
��
X1 , X 2 , . . . , X n
�� f ��
�� f �� f �
�� fˆn ��
�� f � �� n ��
��
��
��
�� Fn �� Xk �
Fn : R →
x �→
1
card {k
n
[0, 1]
≤ n, Xk ≤ x}
�Fn ��
�� Lp �
� f − g � Lp =
��
p
R
|f (x) − g(x)| dx
��
�1/p
.
�� f �� Fn�
�� n �� hn �� (hn)n≥1 ��
(hn ) → 0,
(nhn ) → +∞.
��
�� x
Fn (x + hn ) − Fn (x − hn )
.
fˆn (x) =
2hn
�� fˆn �� Fn�
�� fˆn �� fˆn �
�� f ��
��
�
�� fˆ �� ≥ 0� ��
n
n
n
1 �
1 �
1Xk ∈[x±hn ] =
1x∈[Xk ±hn ]
2nhn k=1
2hn k=1
�
n �
1 �
ˆ
fn (x)dx =
1x∈[Xk ±hn ] dx
2nhn k=1
fˆn (x) =
n
1 �
2hn = 1
=
2nhn k=1
��
�� f �� ˆ ��
�� n �� f �� fn�
�� x �� 2nhn × fˆn(x) �
�� 2nh
n
× fˆn (x)
�� [x ± hn]� ��
��
� �
ˆ
2nhn × fn (x) ∼ Binom n,
�
f (t)dt .
[x±hn ]
�� x �� EQ(x) ��
EQ(x) = E
��
��
��
��
fˆn (x) − f (x)
�2 �
.
�
�2
�
�
� �
EQ(x) = E fˆn (x) − f (x) + Var fˆn (x)
�
�
→ f (x).
�� f �� x �� E fˆn(x) n→+∞
� x+hn
�
1
ˆ
f (t)dt → f (x)
E fn (x) =
2hn x−hn
�
�� f �� x ��
�� fˆn �� f ��
�� x
��
fˆn (x) → f (x).
��
�
�
� x+hn
� x+hn
1
f (t)dt 1 −
f (t)dt
4nh2n x−hn
x−hn
1
1
2hn f (x) (1 − 2hn f (x)) ∼ f (x)
.
∼
2
4nhn
2nhn
�
�
ˆ
Var fn (x) =
��
��
��
L
fˆn (x) → f (x)�
2
��
�� f �� x ��
��
fˆn (x) ��
�� fˆn(x)��
��
�� ĝn ��
� �� (Kn)� ��
�� f ��
��
��
K(x) =
�
15
(x
16
0
− 1)2 (x + 1)2
�� − 1 ≤ x ≤ 1,
��
Kn (x) = nα K(nα x),
�� α �� 0 < α < 1� �� K �� Kn ��
��
n
ĝn (x) =
1�
Kn (x − Xk ).
n k=1
�� ĝn �� ĝn �
�� x �→ K (x − X ) �� C
�� |x − Xk | = 1�� ĝn �� C �
1
��
n
k
�� f �� ĝ
1
��
n ��
�� n �� α��
�� EQ(x) ��
��
2
EQ(x) = (E [ĝn (x)] − f (x)) + Var (ĝn (x)) .
�� x
��
E [ĝn (x)] → f (x).
�
�� δ > 0� |y|>δ Kn(y)dy → 0�
��
E [ĝn (x)] = E[Kn (x − X1 )]
�
�
= Kn (x − t)f (t)dt =
=
=
�
�
x+1/nα
x−1/nα
Kn (x − t)f (t)dt
1/nα
−1/nα
nα K(nα u)f (x − u)du
1
−1
K(s)f (x −
s
)ds
nα
→
��
�� x
�
1
K(s)f (x)ds = f (x)
−1
��
ĝn (x) → f (x).
��
��
�
n
Var (ĝn (x)) = Var
=
��
E[Kn (x − X1 ) ] =
2
1�
Kn (x − Xk )
n k=1
�
1
1
Var (Kn (x − X1 )) ≤ E[Kn (x − X1 )2 ].
n
n
�
2
α
Kn (x − t) f (t)dt = [. . . ] ∼ n f (x)
��
��
�
K 2.
��
�� α ��
1� �� ĝn �
��
�� fˆn �� ĝn �� f � ��
��
��
� ĝn − f �L .
�� f ��
��
�� f �� n �� f �� n
�� f � ��
�� fˆn �� ĝn�
� fˆn − f �L1
1
�� n �� f � ��
�� fˆn − f �L �� ĝn − f �L �
1
1

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X1,X2,...,Xn ˆ Fn : R → [0,1] x → 1 n card{k ≤ n, X k ≤ x} f − g Lp = |f

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