Topics in the Theory of Random Noise

The Markovian approximation is not appropriate for real or actual fluctuations.

“It is important to realise that this approximation is not appropriate for real or actual fluctuations, as noted at the inception of the standard theory;”

The assumption that incremental fluctuations are independent is convenient and appropriate for physical systems with Brownian processes but less plausible for biological and other systems where random fluctuations are generated by dynamical systems.

“Although, this is a convenient assumption that is exploited in conventional Bayesian filtering schemes and appropriate for physical systems with Brownian processes, it is less plausible for biological and other systems, where random fluctuations are themselves generated by dynamical systems ([6], p 81).”

Standard Markovian processes have generalised motion with infinite variance and are infinitely jagged or rough, so high-order terms have no precision and can be ignored.

“This is because standard Markovian (c.f., Wiener) processes have generalised motion that has infinite variance and are infinitely ‘jagged’ or rough. This means w′, w″, … and x″, x‴, … have no precision (inverse variance) and can be ignored with impunity.”

Random fluctuations in biophysical systems are almost invariably the product of dynamical systems, which renders their serial correlations analytic.

“This is because random fluctuations in biophysical systems are almost invariably the product of dynamical systems, which renders their serial correlations analytic ([6], p 83; [23]).”