exists a constant such that for all natural numbers . Then all the assumptions (), (), () and (), hence also () are satisfied. In particular, the conditions () and () are satisfied if . the random variables all have the same distribution. Note that the random variables of the sequence don't need to be independent. The interesting point is to admit some dependence between the random number of terms and the sequence . A standard version is to assume (), (), () and the existence of a filtration such that . is a stopping time with respect to the filtration, and . and are independent for every . Then () implies that the event is in , hence by () independent of . This implies (), and together with () it implies (). For convenience (see the proof below using the optional stopping theorem) and to specify the relation of the sequence and the filtration , the following additional assumption is often imposed: . the sequence is adapted to the filtration , meaning the is -measurable for every . Note that () and () together imply that the random variables are independent. Application An application is in actuarial science when considering the total claim amount follows a compound Poisson process within a certain time period, say one year, arising from a random number of individual insurance claims, whose sizes are described by the random variables . Under the above assumptions, Wald's equation can be used to calculate the expected total claim amount when information about the average claim number per year and the average claim size is available. Under stronger assumptions and with more information about the underlying distributions, Panjer's recursion can be used to calculate the distribution of . Examples Example with dependent terms Let be an integrable, -valued random variable, which is independent of the integrable, real-valued random variable with . Define for all . Then assumptions (), (), (), and () with are satisfied, hence also () and (), and Wald's equation applies. If the distribution of is not symmetric, then () does not hold. Note that, when is not almost surely equal to the zero random variable, then () and () cannot hold simultaneously for any filtration , because cannot be independent of itself as is impossible. Example where the number of terms depends on the sequence Let be a sequence of independent, symmetric, and