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Two novel proofs show that the sum of a specific pair of normal random variables is not normal are established in this note. This is one of the most often misunderstood facts by first-year students in probability theory and statistics. The first proof is concise using the moment generating function. The second proof checks whether the moments of the sum have the property of normal distribution. [ABSTRACT FROM AUTHOR]

This note gives some applications of the zeta distribution to number theory. First, a number of probabilistic proofs of the equality between the Riemann zeta functions and some number-theoretic functions are provided. Second, some problems of probability with respect to positive integers considered in the literature are discussed in the zeta distribution case. In contrast to usual heuristic arguments of probabilistic approach in number theory where the uniform distribution is assumed on positive integers, the proofs here are rigorous by utilizing the zeta distributions on positive integers. [ABSTRACT FROM AUTHOR]

This article will be started with a fundamental observation on consecutive non-negative integers. Based on the observation, we will demonstrate that the sequence of non-negative integers is regarded as a uniformly distributed sequence in the ring of the p-adic integers. As a result, the sequence provides us with a faster rate of convergence than the one obtained by simply applying law of the iterated logarithm. [ABSTRACT FROM AUTHOR]

Defines a class of s-dimensional uniformly distributed sequences called the generalized van der Corput sequence. Introduction of the generalized number system based on an expanding integer matrix; Application to numerical integration problems; Sample equation.

The self-similarity in space and time (hereafter self-similarity), either deterministic or statistical, is characterized by similarity exponents and a function of scaled variable, called the scaling function. In the present paper, we address mainly the self-similarity in the limit of early stage, as opposed to the latter one, and also consider the scaling functions that decay or grow algebraically, as opposed to the rapidly decaying functions such as Gaussian or error function. In particular, in the case of simple diffusion, our symmetry analysis shows a mathematical mechanism by which the rapidly decaying scaling functions are generated by other polynomial scaling functions. While the former is adapted to the self-similarity in the late-stage processes, the latter is adapted to the early stages. This paper sheds some light on the internal structure of the family of self-similarities generated by a simple diffusion equation. Then, we present an example of self-similarity for the late stage whose scaling function has power-law tail, and also several cases of self-similarity for the early stages. These examples show the utility of self-similarity to a wider range of phenomena other than the late stage behaviors with rapidly decaying scaling functions. [ABSTRACT FROM AUTHOR]