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.