Domains of Quasi Attraction: Why Stable Processes Are Observed in Reality?

Abstract

From the very start of modelling with power-tail distributions, concerns were expressed about the actual applicability of distributions with infinite expectations to real-world distributions, which usually have bounded ranges. Here, we suggest resolving this issue by shifting the analysis from the true convergence in various CLTs to some kind of quasi convergence, where a stable approximation to, say, normalised sums of n i.i.d. random variables (or more generally, in a functional setting, to the processes of random walks), holds for large n, but not “too large” n. If the range of “large n” includes all imaginable applications, the approximation is practically indistinguishable from the true limit. This approach allows us to justify a stable approximation to random walks with bounded jumps and, moreover, it leads to some kind of cascading (quasi) asymptotics, where for different ranges of a small parameter, one can have different stable or light-tail approximations. The author believes that this development might be relevant to all applications of stable laws (and thus of fractional equations), say, in Earth systems, astrophysics, biological transport and finances.

Keywords

• rates of quasi-convergence
• domain of quasi attraction
• functional limit theorem with stable processes
• Wasserstein distances
• smooth Wasserstein distances
• Kolmogorov’s distance