Universal convergence toward an exponential law and geometric invariance of distributions via a quasi-reversible non-iterative transformation

We introduce a reversible, non-iterative transformation applied to signals with a wide variety of statistical distributions, including thirty-five general distributions as well as spectral distributions arising from random matrix theory (GUE, GOE, GSE, Wigner’s semicircle law, and the Wigner–Dyson law).In all cases studied, the transformation induces convergence toward an exponential distribution with a unit coefficient of variation, independently of the initial distribution. Applying the inverse transformation makes it possible to reconstruct the original signal with small numerical deviations.The introduction of a single control parameter then allows the coefficient of variation to be varied over a wide range while preserving the shape of the reconstructed distribution, thereby defining a geometric invariance. The results are obtained on signals containing up to points, with computation times of less than one second.