Statistical independence in mathematics–the key to a Gaussian law

In this manuscript we discuss the notion of (statistical) independence embedded in its historical context. We focus in particular on its appearance and role in number theory, concomitantly exploring the intimate connection of independence and the famous Gaussian law of errors. As we shall see, this at times requires us to go adrift from the celebrated Kolmogorov axioms, which give the appearance of being ultimate ever since they have been introduced in the 1930s. While these insights are known to many a mathematician, we feel it is time for both a reminder and renewed awareness. Among other things, we present the independence of the coefficients in a binary expansion together with a central limit theorem for the sum-of-digits function as well as the independence of divisibility by primes and the resulting, famous central limit theorem of Paul Erdős and Mark Kac on the number of different prime factors of a number $$n\in{\mathbb{N}}$$ n ∈ N . We shall also present some of the (modern) developments in the framework of lacunary series that have its origin in a work of Raphaël Salem and Antoni Zygmund.

When Intrinsic Randomness Could Come From the Finite/Infinite Transition

Quantum physics is non-causal, and randomness is so-called “intrinsic”. We propose no less than an 18th interpretation of it through non-Archimedean geometry to bring back causality, respect of the Kolmogorov axioms and the existence of hidden variables. For these latter ones, we show that they cannot be in any Hilbert space and hence could not be detected in any traditional experiment. We end through proposing two experiments which would prove the non-Archimedean nature of our universe. The first one consists in a new disruptive type of quantum radar. The second one explains how viscosity naturally occurs in fluid mechanics whereas Boltzmann’s approach only considers elastic shocks at the molecular scale.

Equations of quantum theory in the space of randomly joint quantum events

The dynamics of the system in the space of random joint events is considered. The symmetric difference of events is introduced in space based on the Kolmogorov axioms. To describe quantum effects in the dynamics of the system, an additional axiom is introduced for random joint events: “the symmetric sum of random events.” In the generated space of random joint events, an equation is constructed for the probability of a system transition between two events. It is shown that for pairwise joint events it is equivalent to the equation of quantum mechanics.

Analytical justification of formulas for calculating the probability of failure-free operation for simple reliability models

In the majority of works connected with research of reliability of production and other technical systems the Kolmogorov axiomatics is applied. The corresponding analytical apparatus allows to solve problems of maximization of probability of performance of production tasks (for example, daily schedule of steel smelting or monthly production plan). Evaluation of reliability of both production and information systems is a mandatory procedure in their design and is based on the construction of a special structural scheme, called the system model for calculating its reliability. It should be noted that Kolmogorov’s axioms are formulated for a system consisting of two elements. The corresponding graphical explanation of axiom equity is based on an imaginary experiment on a random drop of a certain point on a unit square, inside of which there are two intersectingcircles whose planes are proportional to the probability of reliable operation of the corresponding elements. But there is no mention that the centers of circles should be at a certain distance from each other. In this paper, for the known Kolmogorov axioms underlying the classical probability theory, their analytical conclusion is given.

On the Nature of Logic Trees in Probabilistic Seismic Hazard Assessment

An objection sometimes made against treating the weights of logic tree branches as probabilities relates to the Kolmogorov axioms, but these are only an obstacle if one believes that logic tree branches represent a seismic source model or ground motion model as being “true.” Models are never true, but some models are better than others. It is argued here that a logic tree weight represents the probability that the model in question is better than the others considered. Only one branch can be the best one, and one branch must be the best one. It is also argued that there are situations in PSHA where uncertainty exists but the analyst lacks the means to express it. Therefore it is not necessarily the case that more information increases uncertainty; it may be that more information increases the possibility of expressing uncertainty that was previously unmanageable.