40 1. Random Walk and Discrete Heat Equation
To do this, assume #(Zd\A) = K, let
˜
T
A
= min{n 1 : Sn A},
and for x, y Zd \ A, we define
J(x, y) = P{
˜
T
A
∞,S
˜A
T
= y | S0 = x}.
Then J is a K × K matrix. In fact (Exercise 1.25),
(1.31) (J I) G = −I.
In particular, G is invertible.
1.6. Exercises
Exercise 1.1. Suppose that X1,X2,... are independent, identically
distributed random variables such that
E[Xj] = 0, P{|Xj| K} = 0,
for some K ∞.
Let M(t) =
E[etXj
] denote the moment generating function
of Xj. Show that for every t 0, 0,
P{X1 + · · · + Xn n} [M(t)
e−t]n.
Show that for each 0, there is a t 0 such that
M(t) e−t 1. Conclude the following: for every 0,
there is a ρ = ρ( ) 1 such that for all n,
P{|X1 + · · · + Xn| n} 2
ρn.
Show that we can prove the last result with the boundedness
assumption replaced by the following: there exists a δ 0
such that for all |t| δ,
E[etXj
] ∞.
Exercise 1.2. Prove the following: there is a constant γ (called
Euler’s constant) and a c such that for all positive integers n,


n
j=1
1
j


γ log n
c
n
.
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