Wald's Equation - Statement

Statement

Let (Xn)n ∈ N be an infinite sequence of real-valued random variables and let N be a nonnegative integer-valued random variable. Assume that

1. (Xn)n ∈ N are all integrable (finite-mean) random variables,
2. E = E P(N ≥ n) for every natural number n, and
3. the infinite series satisfies

Then the random sums

are integrable and

If, in addition,

4. (Xn)n ∈ N all have the same expectation, and
5. N has finite expectation,

then

Remark: Usually, the name Wald's equation refers to this last equality.

Read more about this topic:  Wald's Equation

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