Wald's Equation - Statement

Statement

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

1. (X_n)_{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. (X_n)_{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.