
Lecture 5
Limit of empirical process:
Brownian bridge. Distribution of
χ
2
goodness of fit statistic
Let us start by introducing a generalization of Brownian motion. Let
F(x) be a continuous distribution function; and let w(t), 0 ≤ t ≤ 1,
denote a standard Brownian motion. Then
W (x) = w ◦F(x) = w(F(x))
is called a Brownian motion with respect to time F(x). In more detail,
the standard Brownian motion on [0,1] is a zero-mean Gaussian pro-
cess with independent increments. That is, for any k and any collection
of k points 0 = t
0
< t
1
< ··· < t
k
< t
k+1
= 1 the increments
∆w(t
j
) = w(t
j+1
) −w(t
j
), j = 0, ..., k,
are independent Gaussian random variables. The distribution ...