The Gaussian distribution revisited

A new construction of the Gaussian distribution is introduced and proven. The procedure consists of usng fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval with a continuous cumulative distribution. And let be a deterministic continuous fractal interpolating funtion, as introduced by Barnsley, with parameters and fractal dimension for its graph D. Then, the derived measure tends to a Gaussian for all parmeters such that , for all X. This result 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 measure via a single nearly-plane filling fractal interpolator.