Sum Theorem Research Articles

Non-uniform Bounds and Edgeworth Expansions in Self-normalized Limit Theorems

We study Edgeworth expansions in limit theorems for self-normalized sums. Non-uniform bounds for expansions in the central limit theorem are established while imposing only minimal moment conditions. Within this result, we address the case of non-integer moments leading to a reduced remainder. Furthermore, we provide non-uniform bounds for expansions in local limit theorems. The enhanced tail accuracy of our non-uniform bounds allows for deriving an Edgeworth-type expansion in the entropic central limit theorem as well as a central limit theorem in total variation distance for self-normalized sums.

Some properties of discrete classes of weights satisfying reverse Hölder inequality and reverse inequality for nonincreasing sequences

Abstract In this paper, we prove some new discrete inequalities with weights of Hardy’s type and employ them to establish some properties of discrete weighted Gehring class of sequences satisfying reverse Hölder inequality. We also consider the weighted class of sequences satisfying a reverse inequality for nonincreasing sequences and prove some higher summability theorems.

Convergence rates in the limit theorems for random sums of [formula omitted]-orthogonal random variables

In the paper, upper bounds for the convergence rate in the limit theorems for random sums of m-orthogonal random variables are estimated using the K-functional method. Our results are extensions of some known results related to random sums.

On the local limit theorems for linear sequences of lower psi-mixing Markov chains

In this paper we investigate the local limit theorem for partial sums of linear sequences of the form Xj=∑i∈Zaiξj−i. Here (ai)i∈Z is a sequence of constants satisfying ∑i∈Zai2<∞ and (ξi)i∈Z are functions of a stationary Markov chain with mean zero and finite second moment. The Markov chain is assumed to satisfy one-sided lower psi-mixing condition.

Solution Representations for Poisson’s Equation, Martingale Structure, and the Markov Chain Central Limit Theorem

The solution of Poisson’s equation plays a key role in constructing the martingale through which sums of Markov correlated random variables can be analyzed. In this paper, we study three different representations for the solution for countable state space irreducible Markov chains, two based on entry time expectations, and the other based on a potential kernel. Our consideration of null recurrent chains allows us to extend our theory to positive recurrent nonexplosive Markov jump processes. We also develop the martingale structure induced by these solutions to Poisson’s equation, under minimal conditions, and establish verifiable Lyapunov conditions to support our theory. Finally, we provide a central limit theorem for Markov dependent sums, under conditions weaker than have previously appeared in the literature.

Three conjectures about character sums

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya–Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess’ estimate for short character sums, and upper bounds for L(1,chi ) and L(1+it,chi )) are more-or-less “equivalent”. We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions.

Multivariate central limit theorems for random clique complexes

Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: the vector of critical simplex counts attained by a lexicographical Morse matching,the vector of simplex counts in the link of a fixed simplex, andthe vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U-statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein’s method.

Exact evaluations and reciprocity theorems for finite trigonometric sums

Exact evaluations and reciprocity theorems for finite trigonometric sums

Unbounded Versions of Two Old Summability Theorems

In this note, we obtain extensions of a theorem of Meyer-König and Zeller and a theorem of Wilansky in that the given results do not require a summability matrix to be a bounded operator from the convergent sequences into themselves. The culmination of the results in this note is that a triangle matrix method T with null columns maps a bounded divergent sequence to a null sequence if and only if the range of T is not closed in the null sequences.

Central Sets Theorem in arbitrary semigroup

In this article we provide a version of the Central Sets Theorem, which was originally introduced by H. Furstenberg and considered to be a joint extension of Finite Sum Theorem and van der Waerden's theorem. The original version of this theorem uses nitely many sequences of functions, but here we will provide a version which uses matrices instead of taking sequences.

An Error Term in the Central Limit Theorem for Sums of Discrete Random Variables

Abstract We consider sums of independent identically distributed random variables whose distributions have $d+1$ atoms. Such distributions never admit an Edgeworth expansion of order $d$, but we show that for almost all parameters the Edgeworth expansion of order $d-1$ is valid and the error of the order $d-1$ Edgeworth expansion is typically of order $n^{-d/2}.$

Estimates of the convergence rate in a limit theorem for geometric sums and some of their applications

Estimates of the convergence rate in a limit theorem for geometric sums and some of their applications

A homeomorphism theorem for sums of translates

For a fixed positive integer n consider continuous functions K1,⋯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_1,\\dots $$\\end{document}, Kn:[-1,1]→R∪{-∞}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ K_n:[-1,1]\\rightarrow {{\\mathbb {R}}}\\cup \\{-\\infty \\}$$\\end{document} that are concave and real valued on [-1,0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-1,0)$$\\end{document} and on (0, 1], and satisfy Kj(0)=-∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_j(0)=-\\infty $$\\end{document}. Moreover, let J:[0,1]→R∪{-∞}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J:[0,1]\\rightarrow {{\\mathbb {R}}}\\cup \\{-\\infty \\}$$\\end{document} be upper bounded and such that [0,1]\\J-1({-∞})\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[0,1]\\setminus J^{-1}(\\{-\\infty \\})$$\\end{document} has at least n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+1$$\\end{document} elements, but it is arbitrary otherwise. For x0:=0<x1<⋯<xn≤xn+1:=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_0:=0<x_1<\\dots < x_n \\le x_{n+1}:=1$$\\end{document}, so called nodes, and for t∈[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in [0,1]$$\\end{document} consider the sum of translates function F(x1,…,xn,t):=J(t)+∑j=1nKj(t-xj)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(x_1,\\ldots ,x_n,t):=J(t)+\\sum _{j=1}^n K_j(t-x_j)$$\\end{document}, and the vector of interval maximum values mj:=mj(x1,…,xn):=maxt∈[xj,xj+1]F(x1,…,xn,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_j:=m_j(x_1,\\ldots ,x_n):=\\max _{t\\in [x_j,x_{j+1}]}F(x_1,\\ldots ,x_n,t)$$\\end{document} (j=0,1,…,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$j=0,1,\\ldots ,n$$\\end{document}). We describe the structure of the arising interval maxima as the nodes run over the n-dimensional simplex. Applications presented here range from abstract moving node Hermite–Fejér interpolation for generalized algebraic and trigonometric polynomials via Bojanov’s problem to more abstract results of interpolation theoretic flavour.

Quenched law of large numbers and quenched central limit theorem for multiplayer leagues with ergodic strengths

We propose and study a new model for competitions, specifically sports multi-player leagues where the initial strengths of the teams are independent i.i.d. random variables that evolve during different days of the league according to independent ergodic processes. The result of each match is random: the probability that a team wins against another team is determined by a function of the strengths of the two teams in the day the match is played. Our model generalizes some previous models studied in the physical and mathematical literature and is defined in terms of different parameters that can be statistically calibrated. We prove a quenched—conditioning on the initial strengths of the teams—law of large numbers and a quenched central limit theorem for the number of victories of a team according to its initial strength. To obtain our results, we prove a theorem of independent interest. For a stationary process ξ=(ξi)i∈Z>0 satisfying a mixing condition and an independent sequence of i.i.d. random variables (si)i∈Z>0, we prove a quenched—conditioning on (si)i∈Z>0—central limit theorem for sums of the form ∑ i=1ng(ξi,si), where g is a bounded measurable function. We highlight that the random variables g(ξi,si) are not stationary conditioning on (si)i∈Z >0.

The Hsu–Robbins–Erdös theorem for the maximum partial sums of quadruplewise independent random variables

Etemadi (1981) [10] and Rio (1995) [27] provided proofs of the Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers under optimal moment conditions without using the Kolmogorov-type maximal inequalities. While this famous result holds for sequences of pairwise independent identically distributed real-valued random variables, a closely related result, the Hsu–Robbins–Erdös strong law of large numbers may fail if the underlying random variables are only assumed to be pairwise independent identically distributed. This note develops Rio's method and uses an approximation technique to establish the Hsu–Robbins–Erdös strong law of large numbers for the maximum partial sums of quadruplewise independent identically distributed random variables. We consider random variables taking values in a real separable Banach space X, but the main result is new even when X is the real line. Previous contributions so far considered the complete convergence of the partial sums or restricted to dependence structures satisfying a Kolmogorov-type maximal inequality.

A Berry–Esseen theorem and Edgeworth expansions for uniformly elliptic inhomogeneous Markov chains

We prove a Berry–Esseen theorem and Edgeworth expansions for partial sums of the form \(\displaystyle S_N=\sum \nolimits _{n=1}^{N}f_n(X_n,X_{n+1})\), where \(\{X_n\}\) is a uniformly elliptic inhomogeneous Markov chain and \(\{f_n\}\) is a sequence of uniformly bounded functions. The Berry–Esseen theorem holds without additional assumptions, while expansions of order 1 hold when \(\{f_n\}\) is irreducible, which is an optimal condition. For higher order expansions, we then focus on two situations. The first is when the essential supremum of \(f_n\) is of order \(O(n^{-{\beta }})\) for some \({\beta }\in (0,1/2)\). In this case it turns out that expansions of any order \(r<\frac{1}{1-2{\beta }}\) hold, and this condition is optimal. The second case is uniformly elliptic chains on a compact Riemannian manifold. When \(f_n\) are uniformly Lipschitz continuous we show that \(S_N\) admits expansions of all orders. When \(f_n\) are uniformly Hölder continuous with some exponent \({\alpha }\in (0,1)\), we show that \(S_N\) admits expansions of all orders \(r<\frac{1+{\alpha }}{1-{\alpha }}\). For Hölder continues functions with \({\alpha }<1\) our results are new also for uniformly elliptic homogeneous Markov chains and a single functional \(f=f_n\). In fact, we show that the condition \(r<\frac{1+{\alpha }}{1-{\alpha }}\) is optimal even in the homogeneous case.

Weighted (Eλ, q)(Cλ, 1) Statistical Convergence and Some Results Related to This Type of Convergence

In this paper, we defined weighted (Eλ,q)(Cλ,1) statistical convergence. We also proved some properties of this type of statistical convergence by applying (Eλ,q)(Cλ,1) summability method. Moreover, we used (Eλ,q)(Cλ,1) summability theorem to prove Korovkin’s type approximation theorem for functions on general and symmetric intervals. We also investigated some of the results of the rate of weighted (Eλ,q)(Cλ,1) statistical convergence and studied some sequences spaces defined by Orlicz functions.

Estimates of the Rate of Convergence in the Limit Theorem for Negative Binomial Random Sums

Estimates of the Rate of Convergence in the Limit Theorem for Negative Binomial Random Sums

A Refinement of Non-Uniform Estimates of the Rate of Convergence in the Central Limit Theorem Under the Existence of Moments of Orders No Higher Than Two

Non-uniform estimates of the rate of convergence in the central limit theorem for sums of independent identically distributed summands are refined under the condition that the summands possess moments of orders no higher than two.

Finiteness theorems for universal sums of squares of almost primes

In this paper we study diagonal quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway–Schneeberger 15 theorem.