
38 Deviation of
b
F
n
(x) from F(x) as a random process
The random variables (4.7) are also dependent random variables
and connected by the functional relationship
k
∑
j=0
∆[
b
F
n
(x
j
) −F(x
j
)] = 0. (4.9)
From (4.8) it easily follows that
E
√
n∆[
b
F
n
(x
j
) −F(x
j
)] ·
√
n∆[
b
F
n
(x
l
) −F(x
l
)]
= ∆F(x
j
)δ
jl
−∆F(x
j
)∆F(x
l
),
so that altogether
the covariance matrix of the random vector (4.7) has the form:
C =
∆F(x
0
) 0
.
.
.
0 ∆F(x
k
)
−
∆F(x
0
)
.
.
.
∆F(x
k
)
(∆F(x
0
),... ,∆F(x
k
)).
(4.10)
We will now establish the following fact, the central limit theorem,
concerning the vector (4.7):
let us denote by Φ(λ ; C) a multidimensional normal distribution
with mean vector 0 and covariance matrix