The Proof and Illustration of the Central Limit Theorem by Brownian Numerical Experiments in Real Time within the Java Applet

We developed the interactive Java applet which made it possible to prove and illustrate by Brownian numerical experiments the Central Limit Theorem (CLT); the theorem which yet constitutes the basis for the Gaussian stochastic processes and the reference case for the non-Gaussian ones. Our approach emphasizes the contrast between theoretical complexity and simplicity provided by our probabilistic simulations. We argue that the present approach should be the preliminary stage for the advanced educational process before the analytical stage is developed. We stress that the Gaussian probability distribution function (PDF) is a stable as distinguished, e.g., from the delta-Dirac, Poisson and t-Student ones which were also considered for comparison. The latter distribution was chosen so as to have all the moments higher than the second order diverging so as to verify the validity of CLT. As our experiments were performed in real time we were able to visualize the convergence of the processes (versus the size of the statistical ensemble of Brownian experiments) both to the variance of the sum of independent identically distributed random variables (which linearly increases with the number of the latter) and to the Gaussian PDF (which is the asymptotic distribution of this sum independently of the single-step PDF used here). We hope that our experimental approach will inspire students to undertake their own studies, e.g., to consider the non-Gaussian processes where CLT is violated which is the modern trend in statistical physics and its applications.