Multivalued regularly varying functions and their applications to limit theorems for unions of random sets

1993 ◽  
pp. 85-99
Author(s):  
Ilya S. Molchanov
Keyword(s):  
Limit Theorems ◽  
Random Sets ◽  
Regularly Varying Functions ◽  
Regularly Varying

Pseudo-Regularly Varying Functions and Generalized Renewal Processes

2018 ◽  
Author(s):  
Valeriĭ V. Buldygin ◽  
Karl-Heinz Indlekofer ◽  
Oleg I. Klesov ◽  
Josef G. Steinebach
Keyword(s):  
Renewal Processes ◽  
Regularly Varying Functions ◽  
Regularly Varying

A Multiplicative Analogue of Schur's Tauberian Theorem

2003 ◽  
Vol 46 (3) ◽  
pp. 473-480 ◽  
Author(s):  
Karen Yeats
Keyword(s):  
Power Series ◽  
Asymptotic Behaviour ◽  
Dirichlet Series ◽  
Tauberian Theorem ◽  
Partial Sums ◽  
Regularly Varying Functions ◽  
Regularly Varying

AbstractA theorem concerning the asymptotic behaviour of partial sums of the coefficients of products of Dirichlet series is proved using properties of regularly varying functions. This theorem is a multiplicative analogue of Schur's Tauberian theorem for power series.

On strongly decreasing solutions of cyclic systems of second-order nonlinear differential equations

2015 ◽  
Vol 145 (5) ◽  
pp. 1007-1028 ◽  
Author(s):  
Jaroslav Jaroš ◽  
Kusano Takaŝi
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Nonlinear Differential Equations ◽  
Radial Symmetry ◽  
Second Order ◽  
Accurate Information ◽  
Regularly Varying Functions ◽  
Cyclic System ◽  
Regularly Varying ◽  
Image Position

The n-dimensional cyclic system of second-order nonlinear differential equationsis analysed in the framework of regular variation. Under the assumption that αi and βi are positive constants such that α1 … αn > β1 … βn and pi and qi are regularly varying functions, it is shown that the situation in which the system possesses decreasing regularly varying solutions of negative indices can be completely characterized, and moreover that the asymptotic behaviour of such solutions is governed by a unique formula describing their order of decay precisely. Examples are presented to demonstrate that the main results for the system can be applied effectively to some classes of partial differential equations with radial symmetry to provide new accurate information about the existence and the asymptotic behaviour of their radial positive strongly decreasing solutions.

Limit theorems for convex hulls of random sets

1993 ◽  
Vol 25 (02) ◽  
pp. 395-414 ◽  
Author(s):  
Ilya S. Molchanov
Keyword(s):  
Limit Theorems ◽  
Random Sets ◽  
Convex Hulls ◽  
Stable Sets ◽  
Limiting Distributions ◽  
Closed Set ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set ◽  
Simple Limit

Let , be i.i.d. random closed sets in . Limit theorems for their normalized convex hulls conv () are proved. The limiting distributions correspond to C-stable random sets. The random closed set A is called C-stable if, for any , the sets anA and conv ( coincide in distribution for certain positive an, compact Kn , and independent copies A 1, …, An of A. The distributions of C-stable sets are characterized via corresponding containment functionals.

scholarly journals Even Order Half-Linear Differential Equations with Regularly Varying Coefficients

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1236
Author(s):  
Vojtěch Růžička
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Type Equation ◽  
Real Numbers ◽  
Regularly Varying Functions ◽  
Varying Coefficients ◽  
Even Order ◽  
Regularly Varying ◽  
Nonoscillation Criterion ◽  
Linear Differential

We establish nonoscillation criterion for the even order half-linear differential equation (−1)nfn(t)Φx(n)(n)+∑l=1n(−1)n−lβn−lfn−l(t)Φx(n−l)(n−l)=0, where β0,β1,…,βn−1 are real numbers, n∈N, Φ(s)=sp−1sgns for s∈R, p∈(1,∞) and fn−l is a regularly varying (at infinity) function of the index α−lp for l=0,1,…,n and α∈R. This equation can be understood as a generalization of the even order Euler type half-linear differential equation. We obtain this Euler type equation by rewriting the equation above as follows: the terms fn(t) and fn−l(t) are replaced by the tα and tα−lp, respectively. Unlike in other texts dealing with the Euler type equation, in this article an approach based on the theory of regularly varying functions is used. We establish a nonoscillation criterion by utilizing the variational technique.

scholarly journals New Classes of Weighted Hölder-Zygmund Spaces and the Wavelet Transform

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Stevan Pilipović ◽  
Dušan Rakić ◽  
Jasson Vindas
Keyword(s):  
Wavelet Transform ◽  
Wavelet Transforms ◽  
Elementary Proof ◽  
Left Inverse ◽  
Regularly Varying Functions ◽  
Continuity Theorem ◽  
New Class ◽  
Zygmund Spaces ◽  
Regularly Varying

We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces𝒮0(ℝn)and𝒮(ℍn+1). We then introduce and study a new class of weighted Hölder-Zygmund spaces, where the weights are regularly varying functions. The analysis of these spaces is carried out via the wavelet transform and generalized Littlewood-Paley pairs.

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