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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