We propose a finite-size scaling analysis method for binary stochastic processes X(t) epsilon {0, 1} based on the second moment correlation length xi for the autocorrelation function C(t). The purpose is to clarify the critical properties and provide a new data analysis method for information cascades. As a simple model to represent the different behaviors of subjects in information cascade experiments, we assume that X(t) is a mixture of an independent random variable that takes 1 with probability q and a random variable that depends on the ratio z of the variables taking 1 among recent r variables. We consider two types of the probability f(z) that the latter takes 1: (i) analog [f(z) = z] and (ii) digital [f(z) = 0(z -1/2)]. We study the universal functions of scaling for xi and the integrated correlation time tau. For finite r, C(t) decays exponentially as a function of t, and there is only one stable renormalization group (RG) fixed point. In the limit r -> infinity, where X(t) depends on all the previous variables, C(t) in model (i) obeys a power law, and the system becomes scale invariant. In model (ii) with q not equal 1/2, there are two stable RG fixed points, which correspond to the ordered and disordered phases of the information cascade phase transition with the critical exponents beta = 1 and nu(parallel to) = 2.
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