The work of Bush and Mosteller  on stochastic learning models leads to certain limiting distributions of response probabilities which are of a somewhat different character from those commonly encountered in mathematical statistics. These authors have derived many useful relations between the moments of the distributions but, in deducing the general nature of the distributions themselves, they have relied chiefly on Monte Carlo methods. In the present paper, we consider the two experimenter-controlled events model and, in the "equal alpha" case, derive limiting distributions for certain special cases by inverting the solution of a moment generating functional equation.
"Limiting Distributions Associated with Certain Stochastic Learning Models." Ann. Math. Statist. 33 (4) 1281 - 1285, December, 1962. https://doi.org/10.1214/aoms/1177704360