The
points out some time index and does so based on the information about the path
of the process up to that time index. For a particular path
one can have
for at most one
index:
For these reasons the
is sometimes called "stopping time".
The
pre
field is a collection of scenarios that lead to the optional event
.
In
particular,
Thus, for every
the time
is defined and then
for
.
In other words,
is the part of the path
after the stopping time
with the
pre
part shifted forward and
away:
Assuming the representation of the formula
(
Random walk space
),
In other words,
is the stopping time calculated from the path
obtained by shifting forward the original path
by
calculated on
:
and we
continue
The
is the path obtained by shifting forward
by the
calculated on
We call it
:
Hence, we obtain another way to understand
and
.
Proposition
(Maximum of random walk) Let
be a stationary independent process,
is the associated random walk and the r.v.
are defined
by
Then the statements A,B,C are equivalent and the statements a,b,c are
equivalent.
A.
.
B.
.
C.
.
a.
.
b.
.
c.
.
Proof
Note that the events
and
are permutable events. Therefore, according to the proposition
(
Hewitt and Savage zeroorone
law
) the values 1,0 are the only possible values for
and
.
Hence, if we prove equivalence of A,B,C then we also obtain equivalence of
a,b,c.
We prove
as follows. According to the proposition
(
BetaK separation of random
walk
), the variables
are iid. Hence, the proposition
(
Strong law of large
numbers for iid r.v.
) applies and we
derive
where
.
We also
have
Hence,
Also,
thus
This implies B.
By definition of
,
hence, B implies C. The C implies A by definitions of
and
.
Proposition
(Eventuality of random walk) Let
be a stationary independent process and
is the associated random walk. There are only four mutually exclusive
possibilities, each taking place a.s.
1.
,
2.
,
3.
,
4.
.
