Random walk.

e consider the sequence of r.v. and a triple . The $\Omega$ is the event space of and $\QTR{cal}{F}$ is the minimal $\sigma$ -algebra that makes all measurable. We think of $X_{n}$ as a consecutive sequence of trials as increases. Hence, is a "process" and is a time parameter.

We use the notation $\QTR{cal}{F}_{n}$ to represent the $\sigma$ -algebra containing the information available up to time . Hence, $\QTR{cal}{F}_{n}$ is the minimal $\sigma$ -algebra that makes the family measurable.

We use the notation to represent the $\sigma$ -algebra containing the information that comes after the time . Hence, is the minimal $\sigma$ -algebra that makes the the family measurable.

We use the notation to represent the minimal $\sigma$ -algebra that contains all of the $\QTR{cal}{F}_{n}$ , . By the minimality of $\QTR{cal}{F}$ we have MATH

Proposition

(Random walk space approximation). Given $\varepsilon>0$ and there exists such that MATH

We use the notation for the intersection

(Remote field)

Sometimes it is convenient to think of $\Omega\,$ as the product space

(Random walk space)

where each $\Omega_{n}$ is the probability space for $X_{n}$ . The

is the product measure consistent with the d.f. $F_{n}$ on each $\Omega_{n}$ . The $\Omega$ is a collection of infinite sequences of real numbers: MATH

The $\tau$ denotes the "shift":

(Shift)

The $\tau^{k}$ is the $\tau$ applied

times.

The denotes the set of permutations of integers from the range . A permutation (and similarly ) produces a mapping on and on $\Omega$ according to the rules MATH

(Permutation)

Definition

(Invariant set) The set is called "invariant" if .

Definition

(Permutable set) The set is called "permutable" if $\pi\Lambda=\Lambda$ for every permutation of finite number of positions. A function is "permutable" if .