PREVIOUS SECTION: Stopping Times - Generalization onto 2-dimensional case

1.9   Stationary Sequences and Processes

Discrete Time

Definition 8

(a) Let   be a sequence of r.v.'s. It is stationary, if  ,  ,    ,     (b) Similarly,   is stationary, if  , the above equality holds.

Continuous Time

Definition 8'

(a) Let   be a sequence of r.v.'s. It is stationary, if  , ,    ,     (b) Similarly,   is stationary, if  , the above equality holds.

Definition 9

A sequence of events   is stationary, if   is stationary.

Assume to be stationary,

 ,  .

Introduce r.v.'s:

Lemma 11

(a)       ; (b)       ; (c)

Proof  of Lemma 11.
(a)

(b)

(c)

QDE

Corollary 2

,   .

Proof.  Note:

  and   are either finite or infinite simultaneously.

QDE

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

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