scholarly journals Statistical independence in mathematics–the key to a Gaussian law

2020 ◽  
Author(s):  
Gunther Leobacher ◽  
Joscha Prochno
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Historical Context ◽  
Lacunary Series ◽  
Statistical Independence ◽  
Binary Expansion ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
Kolmogorov Axioms

Abstract 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.

The Patterns of Large Numbers

2019 ◽  
pp. 43-66
Author(s):  
Steven J. Osterlind
Keyword(s):  
Probability Theory ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
French Revolution ◽  
Central Limit ◽  
Historical Context ◽  
Binomial Theorem ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
First Time

This chapter advances the historical context for quantification by describing the climate of the day—social, cultural, political, and intellectual—as fraught with disquieting influences. Forces leading to the French Revolution were building, and the colonists in America were fighting for secession from England. During this time, three important number theorems came into existence: the binomial theorem, the law of large numbers, and the central limit theorem. Each is described in easy-to-understand language. These are fundamental to how numbers operate in a probability circumstance. Pascal’s triangle is explained as a shortcut solving some binomial expansions, and Jacob Bernoulli’s Ars Conjectandi, which presents the study of measurement “error” for the first time, is discussed. In addition, the central limit theorem is explained in terms of its relevance to probability theory, and its utility today.

Statistics of Hecke eigenvalues for GL(𝑛)

2019 ◽  
Vol 31 (1) ◽  
pp. 167-185
Author(s):  
Yuk-Kam Lau ◽  
Ming Ho Ng ◽  
Yingnan Wang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Covariance Matrix ◽  
Central Limit ◽  
Cusp Forms ◽  
Statistical Independence ◽  
Two Dimensional ◽  
Hecke Operator ◽  
Hecke Eigenvalues ◽  
Compact Region

Abstract A two-dimensional central limit theorem for the eigenvalues of {\mathrm{GL}(n)} Hecke–Maass cusp forms is newly derived. The covariance matrix is diagonal and hence verifies the statistical independence between the real and imaginary parts of the eigenvalues. We also prove a central limit theorem for the number of weighted eigenvalues in a compact region of the complex plane, and evaluate some moments of eigenvalues for the Hecke operator {T_{p}} which reveal interesting interferences.

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

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