Discussion
Let \(\{X_{n}\}\) be a sequence of independent random variables. Define the \(\sigma\)-field :
\[ F = \bigcap_{n=1}^{\infty} \sigma(X_{n}, X_{n+1}, \ldots). \]
Suppose \(A \in f_{e}\). Then, which of the following statements is more appropriate ?
Answer: D
The appropriate statement is P(A) = 0 or P(A) = 1.
F, the intersection of \(\sigma(X_{n}, X_{n+1}, \ldots)\) over all n, is the tail \(\sigma\)-field of the sequence (printed in the question as \(f_{e}\)). Kolmogorov's zero-one law says that when the \(X_{n}\) are independent, every tail event has probability 0 or 1.
Examples of tail events are 'the series \(\sum X_{n}\) converges' and 'the limit superior of \(X_{n}\) exceeds 5'. Each is unaffected by changing finitely many terms, and each is either almost certain or almost impossible.
Saying always 0 or always 1 is too strong, since both values occur for different tail events (A and its complement are both tail events). \(P(A) = P(A^{c})\) would force \(P(A) = 1/2\), which the law excludes.
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