THE GAUSSIAN DISTRIBUTION REVISITED

A new construction of the Gaussian distribution is introduced and prov en. The procedure consists of using fractal interpolating functions, w ith graphs having increasing fractal dimensions, to transform arbitrar y continuous probability measures defined over a closed interval. Spec ifically, let X be any probability measure on the closed interval I wi th a continuous cumulative distribution. And let f(Theta,D):I-->R be a deterministic continuous fractal interpolating function, as introduce d by Barnsley (1986), with parameters Theta and fractal dimension for its graph D. Then, the derived measure Y = f(Theta,D)(X) tends to a Ga ussian for all parameters Theta such that D-->2, for all X. This resul t illustrates that plane-filling fractal interpolating functions are ' intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measu re via a single nearly-plane filling fractal interpolator.