Limiting Distributions of Response Probabilities

Abstract

In an earlier paper [3], one of us considered certain limiting distributions of response probabilities arising from the two experimenter-controlled events learning model of Bush and Mosteller [1]. There, as here, the probabilistic model took the form of a Markov process $p_0, p_1 \cdots$ satisfying the following conditions: (i) $p_0$ has an arbitrary distribution on (0, 1); (ii) if $p_n$ is given, then $p_{n+1} = a_1 + \alpha_1p_n$ with probability $\pi_1$ and $p_{n+1} = a_2 + \alpha_2p_n$ with probability $\pi_1$; (iii) $\pi_1 + \pi_2 = 1, 0 \leqq a_j \leqq 1$ and $0 \leqq \alpha_j \leqq 1 - a_j, (j = 1, 2)$. The random variable $p_n$ is called "the response probability on trial $n$." It has been shown by Karlin [2] that a limiting distribution exists as $n \rightarrow \infty$. If $p$ is the random variable of this limiting distribution, it can be shown ([1], p. 98) that the distribution of $p$ is concentrated on $\lbrack\min (\lambda_1, \lambda_2), \max (\lambda_1, \lambda_2)\rbrack$, where $\lambda_1 = a_1/(1 - \alpha_1)$ and $\lambda_2 = a_2/(1 - \alpha_2)$. In the present note, it is shown that for the case $\alpha_1 = \alpha_2 = \alpha, \pi_1 = \pi_2 = \frac{1}{2}$, the characteristic function of the distribution of the random variable \begin{equation*}\tag{1.1}x = (\lambda_1 + \lambda_2 - 2p)/(\lambda_2 - \lambda_1),\quad (\lambda_1 \neq \lambda_2),\end{equation*} when suitably standardized, tends to the characteristic function of the standardized normal distribution as $\alpha \rightarrow 1$.