Stochastic Processes and Financial Mathematics
(part one)
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8.5 Kolmogorov’s 0-1 law \(\offsyl \)
This section is off-syllabus and is marked with a \(\offsyl \). It contains an intriguing fact about sequences of \(\sigma \)-fields, but it lives somewhere in between the material covered within our own course and
MAS31002/61022 (Probability with Measure). It is mainly of interest to those taking MAS31002/61022 alongside this course, and we will not use it within our course, so it is best placed off-syllabus. It has a close
connection to the second Borel-Cantelli lemma, which is introduced in MAS31002/61022.
Let \((\mc {F}_n)_{n\in \N }\) be a sequence of \(\sigma \)-fields, on the same probability space \((\Omega ,\mc {F},\P )\). The tail \(\sigma \)-field \(\mc {T}\) of \((\mc {F}_n)\) is defined
by
\(\seteqnumber{0}{8.}{1}\)
\begin{equation}
\label {eq:tail_sigma_field} \mc {T}=\bigcap _{n=1}^\infty \sigma (\mc {F}_n,\mc {F}_{n+1},\ldots ).
\end{equation}
Note that \(\mc {T}\) is a \(\sigma \)-field by Lemma 2.1.5. The intuition here is that \(\sigma \)-field \(\mc {T}\)
contains events that depend only on information ‘in the tail’ of the sequence \((\mc {F}_n)\). That is, if we have an event \(E\in \mc {T}\), then for any \(N\in \N \) we could tell whether \(E\) occurred by
looking only at the occurrence of events \(E'\in \mc {F}_n\) for \(n\geq N\). For example, if \((X_n)\) is a sequence of random variables and \(\mc {F}_n=\sigma (X_n)\) then the almost sure
limit \(X_n\stackrel {a.s.}{\to } X\), if it exists, will satisfy \(X\in m\mc {T}\).
The next result may appear surprising at first. The key point is that the \(\mc {F}_n\) are assumed to be independent, which means that they have no information in common. Consequently (8.2) implies that \(\mc {T}\) contains no information.
Proof: Let \(A\in \mc {T}\). Then \(A\in \sigma (\mc {F}_{n+1},\mc {F}_{n+2},\ldots )\) for
all \(n\in \N \). This means that \(A\) is independent of \(\sigma (\mc {F}_1,\ldots ,\mc {F}_n)\), for all \(n\). It follows that \(A\) is independent of \(\sigma (\mc {F}_n\-n\in \N )\). However, from
(8.2) we have that \(\mc {T}\sw \sigma (\mc {F}_n\-n\in \N )\), which means that \(A\) is independent
of \(\mc {T}\). Hence \(A\) is independent of \(A\) (this is not a typo!), which means that \(\P [A]=\P [A\cap A]=\P [A]\P [A]=\P [A]^2\). The only solutions of the equation \(x^2=x\) are \(0\) and \(1\),
hence \(\P [A]=0\) or \(\P [A]=1\). ∎
-
Let us assume the same independence assumption as Theorem 8.5.1, and suppose that \(X\in m\mc {T}\). Then \(\{X\leq x\}\in \mc {T}\) for all \(x\in \R \), so Theorem 8.5.1 gives that \(\P [X\leq x]\) is either \(0\) or \(1\) for each \(x\in \R \). A bit of analysis shows that if we set \(c=\inf \{x\in \R \-\P [X\leq
x]=1\}\) then in fact \(\P [X=c]=1\). Therefore, any random variable that is \(\mc {T}\) measurable is almost surely equal to a constant.