Second-order heavy-tailed distributions and tail analysis

This correspondence studies the second-order distributions of heavy-tail distributed random variables (RVs). Two models for the heavy-tailed distributions are considered: power law and epsi-contaminated distributions. Special cases of the models considered include 1) RVs formed by the product of two independent, but not necessarily identically distributed, heavy-tailed RVs X and Y, such that Z=XY, and 2) RVs formed through squaring a heavy-tail distributed RV X, such that W=X². Tail analysis of the RVs, their cross terms, and square values shows the ordering of their tail heaviness. The following results hold strictly for power law distributions and, under mild conditions, for epsi-contaminated distributions: The tail of f _Z(x) is heavier than that of f_X(x), and the tail of f_W(x) is heavier than that of f_Z(x), where f _(.)(x) denotes the probability density distribution of the corresponding random variable. The heaviness of the tails indicates that robust methods of sample combination and output determination should be utilized to avoid undue influence of outliers and degradation in performance. As examples, the denoising and frequency-selective filtering problems under the derived cross and square statistics for hyperbolic-tailed and epsi-contaminated models are considered. Simulation results indicate that the weighted myriad (WMY) filter outperforms the weighted median (WM) filter, and the WM filter outperforms the weighted sum [finite-impulse-response (FIR)] filter. The results may be exploited in higher order applications of heavy-tailed distributions in networking, such as data traffic modeling, and in nonlinear signal and image processing, such as polynomial and Volterra filtering