Related papers: Tail Probability and Divergent Series
This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…
By a classical principle of analysis, sufficiently thin subsequences of general sequences of functions behave like sequences of independent random variables. This observation not only explains the remarkable properties of lacunary…
The usual nonnegative modulus function is based on addition. A natural different modulus function on the set of positive reals is introduced. Arguments for results for series through the usual modulus function are transformed to arguments…
This paper establishes connections between the boundary behaviour of functions representable as absolutely convergent Dirichlet series in a half-plane and the convergence properties of partial sums of the Dirichlet series on the boundary.…
The divergence of the harmonic series is proved by direct comparison with a series whose nth partial sum telescopes to the natural logarithm of n. The key idea is to apply the classical inequality x>=log(1+x) (valid for x>-1) with x=1/k and…
We consider the theory of algebraically closed fields of characteristic zero with multivalued operations $x\mapsto x^r$ (raising to powers). It is in fact the theory of equations in exponential sums. In an earlier paper we have described…
We study Dirichlet series arising as linear functionals on an inner product space of meromorphic functions and establish a relation between the discontinuities of the former on the boundary and the poles and zeros of the latter on the…
Using an intrinsic approach, we study some properties of random fields which appear as tail fields of regularly varying stationary random fields. The index set is allowed to be a general locally compact Hausdorff Abelian group $\mathbb{G}$.…
We prove that certain quotients of entire functions are characteristic functions. Under some conditions, the probability measure corresponding to a characteristic function of that type has a density which can be expressed as a generalized…
This paper investigates the problem of extending measure theory to non-separable structures, from generalized descriptive set theory to a broader class of spaces beyond this framework. While various notions, such as the ideal of measure…
The aim of this paper is to extend probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive probability for product logic formulas, called…
We derive theorems which outline explicit mechanisms by which anomalous scaling for the probability density function of the sum of many correlated random variables asymptotically prevails. The results characterize general anomalous scaling…
We introduce the class of {\em strongly Rayleigh} probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal…
This paper develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and…
Let $L$ be a countable language. We characterize, in terms of definable closure, those countable theories $\Sigma$ of $\mathcal{L}_{\omega_1, \omega}(L)$ for which there exists an $S_\infty$-invariant probability measure on the collection…
The theory of the functional sequences and series is presented; uniformly convergent, convergent in the sense of a mean square and weakly convergent sequences and series are considered. Sequential approach to constructing generalized…
An interplay between the sum of certain series related to Harmonic numbers and certain finite trigonometric sums is investigated. This allows us to express the sum of these series in terms of the considered trigonometric sums, and permits…
We consider the probability that a weighted sum of $n$ i.i.d. random variables $X_j$, $j = 1, . . ., n$, with stretched exponential tails is larger than its expectation and determine the rate of its decay, under suitable conditions on the…
Suppose we are given two probability measures on the set of one-way infinite finite-alphabet sequences and consider the question when one of the measures predicts the other, that is, when conditional probabilities converge (in a certain…
This paper investigates what can be inferred about an arbitrary continuous probability distribution from a finite sample of $N$ observations drawn from it. The central finding is that the $N$ sorted sample points partition the real line…