1.1. Simple random walk 13
returns to the origin infinitely often. If d 3, the random walk is
transient, i.e., with probability one it returns to the origin only finitely
often. Also,
P{Sn = 0 for all n 0} 0 if d 3.
1.1.6. Notes about probability. We have already implicitly used
some facts about probability. Let us be more explicit about some of
the rules of probability. A sample space or probability space is a set
Ω and events are a collection of subsets of Ω including and Ω. A
probability P is a function from events to [0, 1] satisfying P(Ω) = 1
and the following countable additivity rule:
If E1,E2,... are disjoint (mutually exclusive) events, then

En =

We do not assume that P is defined for every subset of Ω, but we do
assume that the collection of events is closed under countable unions
and “complementation”, i.e., if E1,E2,... are events so are Ej and
Ω \ Ej.

The assumptions about probability are exactly the assumptions used
in measure theory to define a measure. We will not discuss the difficulties
involved in proving such a probability exists. In order to do many things in
probability rigorously, one needs to use the theory of Lebesgue integration. We
will not worry about this in this book.
We do want to discuss one important lemma that probabilists use
all the time. It is very easy, but it has a name. (It is very common for
mathematicians to assign names to lemmas that are used frequently
even if they are very simple—this way one can refer to them easily.)
Lemma 1.3 (Borel-Cantelli Lemma). Suppose E1,E2,... is a collec-
tion of events such that

P(En) ∞.
Then with probability one at most finitely many of the events occur.
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