Self-modulation processes and resulting generic 1/ f fluctuations

Abstract

We analyze high precision data of transaction intervals in a foreign exchange market, and show that it is nicely approximated by a non-stationary Poisson process whose expectation value is given by a moving average of its own trace. Generalizing this result we introduce novel stochastic processes called the self-modulation processes. By the self-modulation effect, clustering occurs automatically resulting in fat-tailed interval distributions including the Zipf's law in an extreme case. We prove rigorously that the corresponding power spectrum follows the 1/ f spectrum.