Human Nature Creates Non-Stationarity

A necessary condition for deriving such reasoning is the stationarity of the stochastic process. This, however, is not so easily ensured. In hydrology, where the assumption of stationarity would seem considerably easier to satisfy, Milly et al. (2008) stated: "Stationarity is dead." The conditions for simple calculations on the map of a stochastic process therefore no longer exist. In the world of financial phenomena, the absence of stationarity is self-evident. This is addressed, in the statistical sense, by Embrechts, Klüppelberg, and Mikosch (1997). Yet this idea can be contested even earlier, at its very foundation. Financial phenomena cannot be stationary. This would stand in contradiction to a natural and unassailable drive on the part of human beings, societies, and entire states — the drive toward development.

The world, it would seem, has for centuries operated precisely on the foundations of such an idea. The very paradigm of continuous GDP growth rules out the identity of phenomena across different moments in time. Nor is it clear whether the known techniques for reducing non-stationarity are capable of taming this undesirable property — undesirable, that is, from the standpoint of statistical methods blind to all else. Indeed, a different question ought to be asked: whether the application of non-stationarity-elimination techniques makes any sense at all. Carrying out procedures of this kind stands, after all, in contradiction to the natural — dynamic, rather than stationary — property of these processes. It might make as much sense as describing an aircraft, its properties, and its operation, while it sits parked on the airport tarmac. The question of generic non-stationarity — or, put differently, the absence of the stationarity required for the application of statistical mechanics — is thus a significant, and difficult to overlook, obstacle to the application of probabilistic methods in the description of financial phenomena.

Unnatural Time-Indexing — Does the Normal Distribution Even Emerge?

There is, moreover, a second problem. It must be borne in mind that statistical properties per se (the properties of distributions, quantile readings, and even stochastic processes and their properties) are not, in themselves, placed on the map of time. A stochastic process may be ordered according to any indexing formula whatsoever — for instance, in spatial order — and the derivation of the normal distribution follows from the summation of random variables (identical and independent — i.i.d.), not from arranging them on the map of time. And whereas, in hydrology, such a placement of phenomena in time is natural — through the adoption of the block maxima method (the modeling of annual maxima) and the placement of phenomena on the astronomical calendar — in the case of socio-economic phenomena, the time index of the variables within the stochastic process is by no means self-evident. The normal distribution, by way of the CLT, emerges as the sum of random variables. A single realization of a random variable, in socio-economic phenomena, need not, however, occupy equal spans of time. At certain moments, events unfold faster; at others, more slowly. And it is precisely this completion of the time variable (e.g., spinning a roulette wheel through to its conclusion, or playing a hand of poker through to its end, where each hand may take a different amount of time) — that is, the completion of the probabilities themselves (the probabilities in a game of poker matter only within the confines of a single hand and the configuration of cards dealt in that hand — time has nothing to do with it) — that is the condition for placing such a random variable (an independent variable) alongside other variables within the stochastic process. Arranging such variables on the map of time — for instance, within artificially constructed time boxes — may instead cause the probabilities from one variable to bleed into another, across such an artificial, time-based division of the variables (that is, a division in which it is time that determines where one variable ends and another begins). It is, moreover, well known that the world is accelerating. The same phenomena now unfold more quickly — they complete themselves faster (the 2023 crisis demonstrated the accelerated completion of a banking panic, driven by 24/7 banking and social media).