On the active periods of nonlinear Shot Noise

We study the temporal ‘on-and-off’ structure of the recently introduced nonlinear Shot Noise model [I. Eliazar, J. Klafter, PNAS 102 (2005) 13779; I. Eliazar, J. Klafter, Nonlinear shot noise: Lévy, Noah, & Joseph, Physica A (2006), to appear], in which: (i) shots of random magnitudes ‘bombard’ a system stochastically in time; (ii) the magnitudes of ‘incoming’ shots decay to zero nonlinearly; and, (iii) the overall effect of the shots on the system is additive. When the shot-inflow and shot-decay satisfy certain conditions, the resulting Shot Noise alternates randomly between ‘active periods’ (in which the noise is ‘on’) and ‘silent periods’ (in which noise is ‘off’). The statistical properties of the ‘active’ and ‘silent’ periods are analyzed, and the resulting ‘on-and-off’ Shot Noise structure is explored. Explicit formulae for means, variances, Laplace transforms, and other statistics are derived in closed-form. Based on this analysis, Shot Noise systems are categorized into three classes: transient, null-recurrent, and recurrent; and, these classes are characterized both analytically and probabilistically. Within the null-recurrent class, systems whose ‘active periods’ are governed by heavy-tailed probability distributions are further characterized.