Previously we derived the probability density function (PDF) of the zero-crossing interval for 1/f^α noise and found that the PDF, L(t) obeys the power law of the form 1/t^c whose exponent c relates to the exponent α of the power spectrum density as c = 3-α when 0 < α < 1 and c = (5 - α)/2 when 1< α < 2. (Proc. SPIE Vol. 5471, p. 29, 2004). This analytical result agreed with numerical experiments by Mingesz et al. (Proc. SPIE Vol. 5110, p.312, 2003) for 0.7 less than or equivalent to α < 2, but not for 0 < α less than or equivalent to 0.7; the experimental PDF deviates from the power law in the latter range. We present here a discretized version of the previous theory by noting that the experimental time interval takes discrete numbers. The present result agrees well with the experiment for the whole range of α and explains the deviation from the power law of PDF in the range of small α.

Date Published: 15 June 2007
PDF: 12 pages
Proc. SPIE 6601, Noise and Stochastics in Complex Systems and Finance, 660112 (15 June 2007); doi: 10.1117/12.724586

Published in SPIE Proceedings Vol. 6601:
Noise and Stochastics in Complex Systems and Finance
János Kertész; Stefan Bornholdt; Rosario N. Mantegna, Editor(s)