The authors derive the conditional expectations for future purchases as given by the Beta Binomial/Negative Binomial distribution. They analyze the special case when these conditional expectations are linear. Link to the article
SCHMITTLEIN, D.C., BEMMAOR, A.C. and MORRISON, D.G. (1985). Technical Note—Why Does the NBD Model Work? Robustness in Representing Product Purchases, Brand Purchases and Imperfectly Recorded Purchases. Marketing Science, 4(3), pp. 255-266.