THE POISSON-DISTRIBUTION ON A FRACTAL

Although the galaxy distribution is observed to be scale invariant at some scales, it cannot be identified to a perfect fractal which is a mathematical idealization. In this letter we illustrate, by a simple example, the fact that discretisation and finite sampling effects modify the expected properties of a fractal-like distribution : sampling an ideal fractal by a finite sequence of independent observations, we obtain a Poisson distribution on a fractal. Analyzing its main properties, with emphasis on number counts probabilities and mean N-points correlation functions, we illustrate these effects. Alhough this is not intended to modelize the galaxy distribution, we point out some similarities which justify the interest of this distribution for tests on statistical methods.