The Schröder equation and its solutions of uniformly regular variation with respect to a given function

2017 ◽  
Vol 24 (5) ◽  
pp. 784-796 ◽  
Author(s):  
Hojjat Farzadfard
Keyword(s):  
Regular Variation ◽  
Schröder Equation ◽  
Uniformly Regular

Stationary measures for multitype branching processes

1975 ◽  
Vol 12 (2) ◽  
pp. 219-227 ◽  
Author(s):  
Fred Hoppe
Keyword(s):  
Functional Equation ◽  
Regular Variation ◽  
Invariant Measures ◽  
Branching Processes ◽  
Stationary Measures ◽  
Multitype Branching Processes ◽  
Watson Process ◽  
Schröder Equation

The multitype Galton-Watson process is considered both with and without immigration. Proofs are given for the existence of invariant measures and their uniqueness is examined by functional equation methods. Theorem 2.1 proves the uniqueness, under certain conditions, of solutions of a multidimensional Schröder equation. Regular variation is shown to play a role in the multitype theory.

Stationary measures for multitype branching processes

1975 ◽  
Vol 12 (02) ◽  
pp. 219-227 ◽  
Author(s):  
Fred Hoppe
Keyword(s):  
Functional Equation ◽  
Regular Variation ◽  
Invariant Measures ◽  
Branching Processes ◽  
Stationary Measures ◽  
Multitype Branching Processes ◽  
Watson Process ◽  
Schröder Equation

The multitype Galton-Watson process is considered both with and without immigration. Proofs are given for the existence of invariant measures and their uniqueness is examined by functional equation methods. Theorem 2.1 proves the uniqueness, under certain conditions, of solutions of a multidimensional Schröder equation. Regular variation is shown to play a role in the multitype theory.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

Point processes, regular variation and weak convergence

1986 ◽  
Vol 18 (01) ◽  
pp. 66-138 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Weak Convergence ◽  
Function Space ◽  
Regular Variation ◽  
Point Processes ◽  
Sufficient Condition ◽  
Space Setting

A method is reviewed for proving weak convergence in a function-space setting when regular variation is a sufficient condition. Point processes and weak convergence techniques involving continuity arguments play a central role. The method is dimensionless and holds computations to a minimum. Many applications of the methods to processes derived from sums and maxima are given.

scholarly journals Characterizations and Examples of Hidden Regular Variation

Extremes ◽  
2004 ◽  
Vol 7 (1) ◽  
pp. 31-67 ◽  
Author(s):  
Krishanu Maulik ◽  
Sidney Resnick
Keyword(s):  
Regular Variation ◽  
Hidden Regular Variation
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