PREVIOUS SECTION: On sigma-algebras, generated by a sequence of r.v.'s., I

(3). A stationary sequence of r.v.'s.

Definition 12

A sequence   (or  ) is stationary, if   (or without ""),   (or  ),

In particular, all   are identically distributed and all finite-dimensional vectors   are i.d. (for a fixed  ).

Examples

1)   - i.i.d.
2)  
3)   ,  

Introduce a shift transformation   on the set of  -measurable (or  -measurable) r.v.'s:

1)    

2) if  , then  

3) if  , then   .

Note:   is measure-preserving:   .

Introduce a shift transformation   on   (or  ):

  iff is measurable iff   is -valued.
Then

  iff

For any  , introduce  

.

In the case of   we can introduce  , too. And - identical transformation.

Definition 13

A  -measurable (or   -...) r.v.   is invariant (w.r.to  ), if   An event   (or  ) is invariant (w.r.to  ), if

Note that   a.s.     ,  

Comments, examples...

Definition 14

A stationary sequence   is ergodic (w.r.to  ), if       (),   is invariant (or is invariant a.s.)

Remark 5

All invariant events (sets) form a  s-algebra   (invariant  s-algebra).

Lemma 16

(1)   (or  ) the sequence of events   (or  ) is stationary; (2) If   is stationary egrodic, then   (or  ) with  holds

Proof.   (1) follows from definitions.

(2) Set   , then

and  

  is invariant

   .

But   .

QDE

Proof.   (a) The case  ; (b) the case  .

1) Set   ,   . Then

and   . But

2) For   , put   .

Note:   and   ,

Set

  and

But   .

QDE

Remark 6

In the case  , the "symmetric" statement is true, too: if   is invariant, then   such that  .

Indeed,   is trivial   if   is invariant,   ,   and  ,    

then .

Remark 7

There exists a number of more weaker conditions (than i.i.d. ones) that imply the "triviality" of the tail s-algebra   and, as a corollary, the ergodicity of a stationary sequence.

For instance, introduce the following "mixing" coefficients:

One can show that if   as   , then   is trivial.

But, in general, there are examples when   is not trivial, but   is (i.e. the sequence is ergodic).

Example    ;

Then

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