ccording to the proposition
(
Infinitely often zeroorone
law
), for a random walk
Proposition
(Structure of recurrent values) The
set
is described by one of the following statements.
1.
.
2.
3.
.
4.
.
Note that
means
and
means
.
Proof
The statement 1 is a direct consequence of the proposition
(
BorelCantelli lemma, part
1
). The statement 2 does not follow from
(
BorelCantelli lemma, part
2
) because
are not independent events. Hence, we proceed with the proof of 2.
By the definition (
Limsup and liminf
for sets
), for a family of sets
then
Thus, for any
,
According to the section
(
Operations on sets and
logical statements
), the statement
means
.
Therefore, we
continue
The above statements within the union are disjoint:
Note that
and
imply
.
Hence,
and the events
and
are
independent:
Note that
is distributed as
:
.
Hence, the second probability is
independent:
If follows that if
then we must have
.
Next, we complete the proof by showing that
for any
.
We introduce the
event
and note that the
events
are disjoint. Hence, we
have
and,
consequently,
We now repeat the argument that lead to
starting from
.
Proposition
(Recurrence result 1) If
in pr. then
