A Central Limit Theorem for Families of Stochastic Processes Indexed by a Small Average Step Size Parameter, and Some Applications to Learning Models

Let θ > 0 be a measure of the average step size of a stochastic process { p n (θ) } n =1 (∞) . Conditions are given under which p n (θ) is approximately normally distributed when n is large and θ is small. This result is applied to a number of learning models where θ is a learning rate parameter and p n (θ) is the probability that the subject makes a certain response on the n th experimental trial. Both linear and stimulus sampling models are considered.