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(θ). Conditions are given under whichp n (θ) is approximately normally distributed whenn is large and ϑ is small. This result is applied to a number of learning models where ϑ is a learning rate parameter andp n (θ) is the probability that the subject makes a certain response on thenth experimental trial. Both linear and stimulus sampling models are considered.