Conditioning on Increasing Sigma-Algebras

Consider a random variable Y defined on the probability space {^, &, P} satis­fying E [|Y |] < to, and let &n be a nondecreasing sequence of sub-a-algebras of &: &n c ■^n+1 C &■ The question I will address is, What is the limit of E[Y|&n] for n ^ to? As will be shown in the next section, the answer to this question is fundamental for time series econometrics.

We have seen in Chapter 1 that the union of a-algebras is not necessarily a a-algebra itself. Thus, UTO=1 &n may not be a a-algebra. Therefore, let

image186(3.23)

that is, &to is the smallest a-algebra containing UTO=1 &n ■ Clearly, &to c & because the latter also contains UTO=1 &n ■

The answer to our question is now as follows:

Theorem 3.12: If Yis measurable &, E[|Y|] < to, and {&n} is a nondecreas­ing sequence of sub-a -algebrasof &, then limn^TO E[Y|&n] = E[Y|&TO] with probability 1, where &TO is defined by (3.23).

This result is usually proved by using martingale theory See Billingsley (1986), Chung (1974), and Chapter 7 in this volume. However, in Appendix

3. A I will provide an alternative proof of Theorem 3.12 that does not require martingale theory.

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