dependent terms. General proof This proof uses only Lebesgue's monotone and dominated convergence theorems. We prove the statement as given above in three steps. Step 1: Integrability of the random sum We first show that the random sum is integrable. Define the partial sums Since takes its values in and since , it follows that The Lebesgue monotone convergence theorem implies that By the triangle inequality, Using this upper estimate and changing the order of summation (which is permitted because all terms are non-negative), we obtain where the second inequality follows using the monotone convergence theorem. By assumption (), the infinite sequence on the right-hand side of () converges, hence is integrable. Step 2: Integrability of the random sum We now show that the random sum is integrable. Define the partial sums of real numbers. Since takes its values in and since , it follows that As in step 1, the Lebesgue monotone convergence theorem implies that By the triangle inequality, Using this upper estimate and changing the order of summation (which is permitted because all terms are non-negative), we obtain By assumption (), Substituting this into () yields which is finite by assumption (), hence is integrable. Step 3: Proof of the identity To prove Wald's equation, we essentially go through the same steps again without the absolute value, making use of the integrability of the random sums and in order to show that they have the same expectation. Using the dominated convergence theorem with dominating random variable and the definition of the partial sum given in (), it follows that Due to the absolute convergence proved in () above using assumption (), we may rearrange the summation and obtain that where we used assumption () and the dominated convergence theorem with dominating random variable for the second equality. Due to assumption () and the σ-additivity of the probability measure, Substituting this result into the previous equation, rearranging the summation (which is permitted due to absolute convergence, see () above), using linearity of expectation and the definition of the partial sum of expectations given in (), By using dominated convergence again with dominating random variable , If assumptions () and () are satisfied, then by linearity of expectation, This completes the proof. Further generalizations Wald's equation can be transferred to -valued random variables by applying the one-dimensional version to every component. If are Bochner-integrable random variables taking values in