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A Graphical Aid for Analyzing Autocorrelated Dynamical Systems

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Figurę 3. The Quincunx. A device typically used to demonstrate how a normal distribution might arise in real processes. The white circle represents a bali to be dropped on the rods, black circles, below. The balls accumulate in the bins at the bottom of the Quincunx. There are often 20 or morę rows of rods.

statisticians for years to demonstrate how one can generate a normal distribution from a large number of independent Bemoulli trials.

In the operation of the Quincunx, a bali is dropped from a fixed location on the top rod. The bali bounces off of each rod down onto a rod below, in each case going to the right or left with equal probability, i.e., the bali makes a simple Bemoulli trial as it hits each rod. The bali finally comes to rest in one of the bins located below the last row of rods. As morę and morę balls accumulate in the bins, an approximation to the histogram of a normal distribution is built up by the balls. The statistical reasoning of the appropriateness of the normal distribution draws directly from the properties of sums of large numbers of identical independent Bemoulli trials.

However if we believe in the Laws of Physics, there are no independent Bemoulli trials! The entire outcome is fiilly determined from the moment the bali is dropped. So where does the normal distribution come from? The answer is from a very dense field of virtual repellers present on the top bar (Russell, 1991). This field of repellers, in the limit, appears to be a fractal Cantor set.

In this case, the physical understanding and the statistical reasoning appear to be somewhat at odds. That the two systems of reasoning correspond