Maybe you could try to group outcome distributions in such a way that the different groups’ sums of outcomes have vanishing “covariance” with the other groups’ outcome sums, and hope you get enough groups left and they satisfy some condition to let you apply something like the LLN or CLT.
Maybe you could try to group outcome distributions in such a way that the different groups’ sums of outcomes have vanishing “covariance” with the other groups’ outcome sums, and hope you get enough groups left and they satisfy some condition to let you apply something like the LLN or CLT.