PREVIOUS SECTION: Probabilistic Metrics
Let be a probability space and be a sequence of r.v.'s, . Denote by the -algebra, generated by :
where =-algebra of Borel sets in .
Then, for , - -algebra, generated by ; i.e.
is a minimal -algebra, so that
Another way of description of is:
is a random vector; . Then
where -algebra of Borel sets in .
Finally, -algebra, generated by the whole sequence .
Good Property: | For all , , ,
such that
|
Let now be an integer-valued r.v.
Definition 5 | Random value is called a stopping
time (ST) with respect to , if ,
(or, equivalently - ). |
Another variant of definition is:
Definition 6 | Random value is a ST, if a family of functions such that:
(or, equivalently - a.s.). |
Examples ...
Assume now that is an i.i.d. sequence, is a ST; . Put
|
Lemma 3 | 1) is an i.i.d.
sequence; 2) ; 3) and a random vector are mutually independent. |
Corollary 1 | and are mutually independent. |
Proof of Lemma 3.
We have to show that Borel sets and ,
|
Indeed, 1), 2) and 3).
First, . Then,
|
In particular, take and for . Then the l.h.s. of = the r.h.s. of = |
2) |
Now, take any and replace in by |
1) |
Finally, take any and and replace in by . |
3) |
QDE
Lemma 4 (Wald identity) |
Assume that and
. Then
|
Proof.
(a) Show that .
Note, that , and
and are independent and are independent
(b) Therefore,
QDE
``Induction..."
Lemma 5 | Let be an i.i.d.
sequence;
Then is a ST w.r. to . |
Proof.
QDE
Let's write
instead of |
Lemma 6 | If is a ST w.r. to
and if , then is a ST w.r. to . |
NEXT SECTION: Generalization onto 2-dimensional case
File translated from TEX by TTH, version 1.58, Htex, and with Natalia Chernova as the TeX2gif-convertor.