On high-order discrete derivatives of stochastic variables

We derive an explicit expression for the probability density function of the mth numerical derivative of a stochastic variable. It is shown that the proposed statistics can analytically be obtained based on the original probability characteristics of the observed signal in a simple manner. We argue that this allows estimating the statistical parameters of the original distribution and further, to simulate the noise contribution in the original stochastic process so that the noise component is statistically indistinguishable from the true contribution of the noise in the originally observed data signal.