8 1. Random Walk and Discrete Heat Equation
Since we expect S2n to be of order

n, let us write an integer j
as j = r

n. Then the right-hand side of (1.6) becomes

2
C0

n
1
r2
n
−n
1 +
r

n


n
r
× 1
r

n


n
−r
1
1 (r2/n)
1/2
.

We are about to use the well-known limit
1 +
a
n
n
−→
ea
n ∞.
In fact, using the Taylor’s series for the logarithm, we get for n
2a2,
log 1 +
a
n
n
= a + O
a2
n
,
which can also be written as
1 +
a
n
n
=
ea
1 +
O(a2/n)
.
As n ∞, the right-hand side of (1.6) is asymptotic to

2
C0

n
er2 e−r2 e−r2
=

2
C0

n
e−j2/n.
For every a b,
(1.7) lim
n→∞
P{a

2n S2n b

2n} = lim
n→∞

2
C0

n
e−j2/n,
where the sum is over all j with a

2n 2j b

2n. The right-
hand side is the Riemann sum approximation of an integral where
the intervals in the sum have length 2/n. Hence, the limit is
b
a
1
C0
e−x2/2
dx.
This limiting distribution must be a probability distribution, so we
can see that

−∞
1
C0
e−x2/2
dx = 1.
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