Bernoulli Process - Finite Vs. Infinite Sequences

Finite Vs. Infinite Sequences

The sigma algebra for a single coin toss is the set

with the probabilities

\begin{align} P(\varnothing) &= 0 \\ P(\{H\}) &= p \\ P(\{T\}) &= 1-p \\ P(\{H,T\}) &= 1 \\ \end{align}

Roughly speaking, the sigma algebra for the one-sided infinite case can be thought of as

\{\varnothing, \{H\}, \{T\}, \{H,T\}\} \,\times\, \{\varnothing, \{H\}, \{T\}, \{H,T\}\} \, \times\cdots

although a more formally correct definition is given below. For this infinite case, consider the two cylinder sets

and

Here, the '*' means 'don't care', and so the cylinder set corresponds to 'flipped tails on the first flip, and don't care about the rest'. The measure of (the probability of ) is

\begin{align} P(C_0) &= P(T,*,*,\cdots) \\ &= P(\{T\})\cdot P(\{H,T\}) \cdot P(\{H,T\}) \cdots \\ &= (1-p) \cdot 1 \cdot 1 \cdots \\ &= 1-p \end{align}

That is, the measure of cylinder set is nothing other than the probability of flipping tails, once. Likewise, one may consider the cylinder set

of flipping tails twice in a row, followed by an infinite sequence of 'don't care's. The measure of this set is again exactly equal to the probability of flipping tails twice, and never flipping again. In essence, a finite sequence of flips corresponds in a one-to-one fashion with a cylinder set taken from the infinite product. The definition of the Bernoulli process does not need any special treatment to distinguish the 'finite case' from the 'infinite case': the mechanics covers both cases equally well.

It should be emphasized that this works because the Bernoulli process was defined this way: the sigma algebra consists of the union of all finite-length (but unbounded!) cylinder sets. Infinite-length strings are explicitly excluded from the construction. Thus, letting be the set of all cylidner sets of length n (and is thus a sigma algebra in and of itself), the sigma algebra describing the Bernoulli process is given by

where this sigma algebra is the middle letter in the Bernoulli process triple

The difference between the above, formal definition, and the somewhat sloppy, informal idea that

\mathcal{F}_{\mathrm{informal}} = \{\varnothing, \{H\}, \{T\}, \{H,T\}\} \,\times\, \{\varnothing, \{H\}, \{T\}, \{H,T\}\} \, \times\cdots

is worth noting. In the formal case, each finite set is endowed with a natural topology, the discrete topology; taking the union preserves this notion. In the informal definition, a question arises: what should the topology be? What could it be? One has the choice of the initial topology and the final topology. The formal definition makes it clear: it's the former, not the latter.

Read more about this topic:  Bernoulli Process

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