Chapter 16
Convergence of Random
Walks
This lecture examines the convergence of random walks to the
Wiener process. This is very important both physically and statis
tically, and illustrates the utility of the theory of Feller processes.
Section 16.1 finds the semigroup of the Wiener process, shows
it satisfies the Feller properties, and finds its generator.
Section 16.2 turns random walks into cadlag processes, and gives
a fairly easy proof that they converge on the Wiener process.
16.1
The Wiener Process is Feller
Recall that the Wiener process
W
(
t
) is defined by starting at the origin, by
independent increments over nonoverlapping intervals, by the Gaussian distri
bution of increments, and by continuity of sample paths (Examples 38 and 78).
The process is homogeneous, and the transition kernels are (Section 11.1)
μ
t
(
w
1
, B
)
=
B
dw
2
1
√
2
π
t
e

(
w
2

w
1
)
2
2
t
(16.1)
dμ
t
(
w
1
, w
2
)
d
λ
=
1
√
2
π
t
e

(
w
2

w
1
)
2
2
t
(16.2)
where the second line gives the density of the transition kernel with respect to
Lebesgue measure.
Since the kernels are known, we can write down the corresponding evolution
operators:
K
t
f
(
w
1
)
=
dw
2
f
(
w
2
)
1
√
2
π
t
e

(
w
2

w
1
)
2
2
t
(16.3)
We saw in Section 11.1 that the kernels have the semigroup property, so the
evolution operators do too.
86
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
CHAPTER 16.
CONVERGENCE OF RANDOM WALKS
87
Let’s check that
{
K
t
}
, t
≥
0 is a Feller semigroup. The first Feller property
is easier to check in its probabilistic form, that, for all
t
,
y
→
x
implies
W
y
(
t
)
d
→
W
x
(
t
). The distribution of
W
x
(
t
) is just
N
(
x, t
), and it is indeed true that
y
→
x
implies
N
(
y, t
)
→
N
(
x, t
).
The second Feller property can be checked in its
semigroup form: as
t
→
0,
μ
t
(
w
1
, B
) approaches
δ
(
w

w
1
), so lim
t
→
0
K
t
f
(
x
) =
f
(
x
).
Thus, the Wiener process is a Feller process.
This implies that it has
cadlag sample paths (Theorem 158), but we already knew that, since we know
it’s continuous. What we did not know was that the Wiener process is not just
Markov but strong Markov, which follows from Theorem 159.
It’s easier to find the generator of
{
K
t
}
, t
≥
0, it will help to rewrite it in
an equivalent form, as
K
t
f
(
w
)
=
E
f
(
w
+
Z
√
t
)
(16.4)
where
Z
is an independent
N
(0
,
1) random variable.
(You should convince
yourself that this is equivalent.) Now let’s pick an
f
∈
C
0
which is also twice
continuously di
ff
erentiable, i.e.,
f
∈
C
0
∩
C
2
. Look at
K
t
f
(
w
)

f
(
w
), and apply
Taylor’s theorem, expanding around
w
:
K
t
f
(
w
)

f
(
w
)
=
E
f
(
w
+
Z
√
t
)

f
(
w
)
(16.5)
=
E
f
(
w
+
Z
√
t
)

f
(
w
)
(16.6)
=
E
Z
√
tf
(
w
) +
1
2
tZ
2
f
(
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '06
 Schalizi
 Normal Distribution, Probability theory, Stochastic process, yn, Random walk, Wiener process

Click to edit the document details