Wald's Equation - Discussion of Assumptions

Discussion of Assumptions

Clearly, assumption (1) is needed to formulate assumption (2) and Wald's equation. Assumption (2) controls the amount of dependence allowed between the sequence (Xn)n∈N and the number N of terms, see the counterexample below for the necessity. Assumption (3) is of more technical nature, implying absolute convergence and therefore allowing arbitrary rearrangement of an infinite series in the proof.

If assumption (5) is satisfied, then assumption (3) can be strengthened to the simpler condition

6. there exists a real constant C such that E ≤ C P(N ≥ n) for all natural numbers n.

Indeed, using assumption (6),

and the last series equals E, which is finite by assumption (5). Therefore, (5) and (6) imply assumption (3).

Assume in addition to (1) and (5) that

7. N is independent of the sequence (Xn)n ∈ N and
8. there exists a constant C such that E ≤ C for all natural numbers n.

Then all the assumptions (1), (2), (5) and (6), hence also (3) are satisfied. In particular, the conditions (4) and (8) are satisfied if

9. the random variables (Xn)n ∈ N all have the same distribution.

Note that the random variables of the sequence (Xn)n ∈ N don't need to be independent.

The interesting point is to admit some dependence between the random number N of terms and the sequence (Xn)n ∈ N. A standard version is to assume (1), (5), (8) and the existence of a filtration (Fn)n ∈ N such that

10. N is a stopping time with respect to the filtration, and
11. Xn and Fn–1 are independent for every n ∈ N.

Then (10) implies that the event {N ≥ n} = {N ≤ n – 1}c is in Fn–1, hence by (11) independent of Xn. This implies (2), and together with (8) it implies (6).

For convenience (see the proof below using the optional stopping theorem) and to specify the relation of the sequence (Xn)n∈N and the filtration (Fn)n∈N0, the following additional assumption is often imposed:

12. the sequence (Xn)n∈N is adapted to the filtration (Fn)n∈N, meaning the Xn is Fn-measurable for every n∈N.

Note that (11) and (12) together imply that the random variables (Xn)n∈N are independent.

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