scholarly journals Local Limit Theorem for the Multiple Power Series Distributions

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2067
Author(s):  
Arsen L. Yakymiv
Keyword(s):  
Limit Theorem ◽  
Power Series ◽  
Local Limit Theorem ◽  
Sufficient Conditions ◽  
Local Limit ◽  
Necessary And Sufficient Conditions ◽  
Multiple Power ◽  
Power Series Distributions ◽  
Necessary And Sufficient ◽  
Multiple Power Series

We study the behavior of multiple power series distributions at the boundary points of their existence. In previous papers, the necessary and sufficient conditions for the integral limit theorem were obtained. Here, the necessary and sufficient conditions for the corresponding local limit theorem are established. This article is dedicated to the memory of my teacher, professor V.M. Zolotarev.

scholarly journals A multidimensional generalization of the Rutishauser $qd$-algorithm

2016 ◽  
Vol 8 (2) ◽  
pp. 230-238 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Multiple Power ◽  
Necessary And Sufficient ◽  
Multiple Power Series

In this paper the regular multidimensional $C$-fraction with independent variables, which is a generalization of regular $C$-fraction, is considered. An algorithm of calculation of the coefficients of the regular multidimensional $C$-fraction with independent variables correspondence to a given formal multiple power series is constructed. Necessary and sufficient conditions of the existence of this algorithm are established. The above mentioned algorithm is a multidimensional generalization of the Rutishauser $qd$-algorithm.

scholarly journals Approximation of functions of several variables by multidimensional $S$-fractions with independent variables

2021 ◽  
Vol 13 (3) ◽  
pp. 592-607
Author(s):  
R.I. Dmytryshyn ◽  
S.V. Sharyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Numerical Experiments ◽  
Sufficient Conditions ◽  
Approximation Of Functions ◽  
Independent Variables ◽  
Several Variables ◽  
Multiple Power ◽  
Necessary And Sufficient ◽  
Multiple Power Series

The paper deals with the problem of approximation of functions of several variables by branched continued fractions. We study the correspondence between formal multiple power series and the so-called "multidimensional $S$-fraction with independent variables". As a result, the necessary and sufficient conditions for the expansion of the formal multiple power series into the corresponding multidimensional $S$-fraction with independent variables have been established. Several numerical experiments show the efficiency, power and feasibility of using the branched continued fractions in order to numerically approximate certain functions of several variables from their formal multiple power series.

On the distribution of multiple power series regularly varying at the boundary point

2019 ◽  
Vol 29 (6) ◽  
pp. 409-421 ◽  
Author(s):  
Arsen L. Yakymiv
Keyword(s):  
Limit Theorem ◽  
Power Series ◽  
Boundary Point ◽  
Local Version ◽  
Random Vectors ◽  
Integral Limit ◽  
Nonnegative Coefficients ◽  
Multiple Power ◽  
Regularly Varying ◽  
Multiple Power Series

Abstract Let B(x) be a multiple power series with nonnegative coefficients which is convergent for all x ∈ (0, 1)n and diverges at the point 1 = (1, …, 1). Random vectors (r.v.)ξx such that ξx has distribution of the power series B(x) type is studied. The integral limit theorem for r.v. ξx as x ↑ 1 is proved under the assumption that B(x) is regularly varying at this point. Also local version of this theorem is obtained under the condition that the coefficients of the series B(x) are one-sided weakly oscillating at infinity.

A non-uniform rate of convergence in a local limit theorem

1978 ◽  
Vol 84 (2) ◽  
pp. 351-359 ◽  
Author(s):  
Sujit K. Basu
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Standard Normal ◽  
The Common ◽  
Necessary And Sufficient

AbstractLet {Xn} be a sequence of iid random variables. If the common charac-teristic function is absolutely integrable in mth power for some integer m ≥ 1, then Zn = n−½(X1 + … + Xn) has a pdf fn for all n ≥ m. Here we give a necessary and sufficient condition for sup as n. → ∞, where φ (x) is the standard normal pdf and M(x) is a non-decreasing function of x ≥ 0 such that M(0) > 0 and M(x)/xδ is non-increasing for 0 < δ ≤ 1.

scholarly journals The Abel-type transformations intoℓ

1999 ◽  
Vol 22 (4) ◽  
pp. 775-784
Author(s):  
Mulatu Lemma
Keyword(s):  
Power Series ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sequence Variant ◽  
Series Method ◽  
Power Series Method ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Type Power

Lettbe a sequence in(0,1)that converges to1, and define the Abel-type matrixAα,tbyank=(k+α     k)tnk+1(1−tn)α+1forα>−1. The matrixAα,tdetermines a sequence-to-sequence variant of the Abel-type power series method of summability introduced by Borwein in [1]. The purpose of this paper is to study these matrices as mappings intoℓ. Necessary and sufficient conditions forAα,tto beℓ-ℓ,G-ℓ, andGw-ℓare established. Also, the strength ofAα,tin theℓ-ℓsetting is investigated.

scholarly journals The necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution

2019 ◽  
Vol 22 (1) ◽  
pp. 143-146
Author(s):  
Tran Loc Hung ◽  
Phan Tri Kien
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Sufficient Conditions ◽  
Laplace Distribution ◽  
Asymptotic Distributions ◽  
Necessary And Sufficient Conditions ◽  
Characteristic Functions ◽  
Necessary And Sufficient ◽  
Geometric Sums ◽  
Risk Processes

The geometric sums have been arisen from the necessity to resolve practical problems in ruin prob- ability, risk processes, queueing theory and reliability models, etc. Up to the present, the results related to geometric sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned domains of geometric attraction are still limitative. The main purpose of this article is to introduce concepts on the domain of geometric attraction of standard Laplace distribution. Using method of characteristic functions, the necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution are shown. In special case, obtained result is a weak limit theorem for geometric sums of independent and identically distributed random variables which has been well-known as the second central limit theorem. Furthermore, based on the obtained results of this paper, the analogous results for the domains of geometric attraction of exponential distribution and Linnik distribution can be established. More generally, we may extend results to the domain of geometric attraction of geometrically strictly stable distributions.      

scholarly journals Locally Restricted Compositions I. Restricted Adjacent Differences

10.37236/1954 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Edward A. Bender ◽  
E. Rodney Canfield
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Local Limit Theorem ◽  
Sufficient Conditions ◽  
Local Limit ◽  
The Other ◽  
Simple Condition ◽  
Suitable Constant ◽  
Distinct Part ◽  
Positive Elements

We study compositions of the integer $n$ in which the first part, successive differences, and the last part are constrained to lie in prescribed sets ${\cal L,D,R}$, respectively. A simple condition on ${\cal D}$ guarantees that the generating function $f(x,{\cal L,D,R})$ has only a simple pole on its circle of convergence and this at $r({\cal D})$, a function independent of ${\cal L}$ and ${\cal R}$. Thus the number of compositions is asymptotic to $Ar({\cal D})^{-n}$ for a suitable constant $A=A({\cal L,D,R})$. We prove a multivariate central and local limit theorem and apply it to various statistics of random locally restricted compositions of $n$, such as number of parts, numbers of parts of given sizes, and number of rises. The first and last parts are shown to have limiting distributions and to be asymptotically independent. If ${\cal D}$ has only finitely many positive elements ${\cal D}^+$, or finitely many negative elements ${\cal D}^-$, then the largest part and number of distinct part sizes are almost surely $\Theta((\log n)^{1/2})$. On the other hand, when both ${\cal D}^+$ and ${\cal D}^-$ have a positive asymptotic lower "local log-density", we prove that the largest part and number of distinct part sizes are almost surely $\Theta(\log n)$, and we give sufficient conditions for the largest part to be almost surely asymptotic to $\log_{1/r({\cal D})}n$.

scholarly journals Convergence criterion for branched contіnued fractions of the special form with positive elements

2017 ◽  
Vol 9 (1) ◽  
pp. 13-21 ◽  
Author(s):  
D.I. Bodnar ◽  
I.B. Bilanyk
Keyword(s):  
Continued Fractions ◽  
Special Form ◽  
Sufficient Conditions ◽  
Convergence Criterion ◽  
Sufficient Condition ◽  
Independent Variables ◽  
Positive Elements ◽  
Multiple Power ◽  
Necessary And Sufficient ◽  
Multiple Power Series

In this paper the problem of convergence of the important type of a multidimensional generalization of continued fractions, the branched continued fractions with independent variables, is considered. This fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. When variables are fixed these fractions are called the branched continued fractions of the special form. Their structure is much simpler then the structure of general branched continued fractions. It has given a possibility to establish the necessary and sufficient conditions of convergence of branched continued fractions of the special form with the positive elements. The received result is the multidimensional analog of Seidel's criterion for the continued fractions. The condition of convergence of investigated fractions is the divergence of series, whose elements are continued fractions. Therefore, the sufficient condition of the convergence of this fraction which has been formulated by the divergence of series composed of partial denominators of this fraction, is established. Using the established criterion and Stieltjes-Vitali Theorem the parabolic theorems of branched continued fractions of the special form with complex elements convergence, is investigated. The sufficient conditions gave a possibility to make the condition of convergence of the branched continued fractions of the special form, whose elements lie in parabolic domains.

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