Regularly varying functions in the theory of simple branching processes

1974 ◽  
Vol 6 (3) ◽  
pp. 408-420 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Invariant Measures ◽  
Function Theory ◽  
Branching Processes ◽  
Regularly Varying Function ◽  
Asymptotic Structure ◽  
Regularly Varying Functions ◽  
Limit Laws ◽  
Intimate Connection ◽  
Similar Fact ◽  
Regularly Varying

It is demonstrated for the non-critical and the explosive cases of the simple Bienaymé-Galton-Watson (B. G. W.) process (with and without immigration) that there exists a natural and intimate connection between regularly varying function theory and the asymptotic structure of the limit laws and corresponding norming constants. A similar fact had been demonstrated in connection with their invariant measures in [22]. This earlier study is complemented here by a similar analysis of the process where immigration occurs only at points of “emptiness” of the B. G. W. process.

Regularly varying functions in the theory of simple branching processes

1974 ◽  
Vol 6 (03) ◽  
pp. 408-420 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Invariant Measures ◽  
Function Theory ◽  
Branching Processes ◽  
Regularly Varying Function ◽  
Asymptotic Structure ◽  
Regularly Varying Functions ◽  
Limit Laws ◽  
Intimate Connection ◽  
Similar Fact ◽  
Regularly Varying

It is demonstrated for the non-critical and the explosive cases of the simple Bienaymé-Galton-Watson (B. G. W.) process (with and without immigration) that there exists a natural and intimate connection between regularly varying function theory and the asymptotic structure of the limit laws and corresponding norming constants. A similar fact had been demonstrated in connection with their invariant measures in [22]. This earlier study is complemented here by a similar analysis of the process where immigration occurs only at points of “emptiness” of the B. G. W. process.

scholarly journals On functions derived from regularly varying functions

1977 ◽  
Vol 23 (4) ◽  
pp. 431-438 ◽  
Author(s):  
Laurens De Haan
Keyword(s):  
Regularly Varying Function ◽  
Regularly Varying Functions ◽  
Regularly Varying ◽  
Stieltjes Transforms

A generalization of Karamata's theorem on integrals of regularly varying functions is proved. Using Laplace-Stieltjes transforms it is shown that any regularly varying function with exponent α (α + 1 ∉ N) is asymptotic to another regularly varying function all of whose derivations are regularly varying.

scholarly journals On a limit structure of the Galton–Watson branching processes with regularly varying generating functions

2019 ◽  
Vol 39 (1) ◽  
pp. 61-73
Author(s):  
Azam A. Imomov
Keyword(s):  
Generating Function ◽  
Discrete Time ◽  
Generating Functions ◽  
Asymptotic Representation ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Regularly Varying Functions ◽  
Finite Moment ◽  
Critical Situation ◽  
Regularly Varying

We investigate limit properties of discrete time branching processes with application of the theory of regularly varying functions in the sense of Karamata. In the critical situation we suppose that the offspring probability generating function has an infinite second moment but its tail regularly varies. In the noncritical case, the finite moment of type Ε[x ln x] is required. The lemma on the asymptotic representation of the generating function of the process and its differential analogue will underlie our conclusions.

scholarly journals Multitype branching processes with inhomogeneous Poisson immigration

2018 ◽  
Vol 50 (A) ◽  
pp. 211-228
Author(s):  
Kosto V. Mitov ◽  
Nikolay M. Yanev ◽  
Ollivier Hyrien
Keyword(s):  
Asymptotic Behaviour ◽  
Branching Processes ◽  
Random Measure ◽  
Poisson Random Measure ◽  
Regularly Varying Function ◽  
Limit Distributions ◽  
Second Moments ◽  
Multitype Branching Processes ◽  
Regularly Varying ◽  
Local Intensity

Abstract In this paper we introduce multitype branching processes with inhomogeneous Poisson immigration, and consider in detail the critical Markov case when the local intensity r(t) of the Poisson random measure is a regularly varying function. Various multitype limit distributions (conditional and unconditional) are obtained depending on the rate at which r(t) changes with time. The asymptotic behaviour of the first and second moments, and the probability of nonextinction are investigated.

scholarly journals On Avakumovic’s theorem for generalized Thomas-Fermi differential equations

2016 ◽  
Vol 99 (113) ◽  
pp. 125-137 ◽  
Author(s):  
Jaroslav Jaros ◽  
Takaŝi Kusano
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Positive Solutions ◽  
Negative Index ◽  
Regularly Varying Function ◽  
Regularly Varying Functions ◽  
Thomas Fermi ◽  
Regularly Varying

For the generalized Thomas-Fermi differential equation (|x?|??1x?)? = q(t)|x|??1x, it is proved that if 1 ? ? < ? and q(t) is a regularly varying function of index ? with ? > ?? ? 1, then all positive solutions that tend to zero as t ? 1 are regularly varying functions of one and the same negative index p and their asymptotic behavior at infinity is governed by the unique definite decay law. Further, an attempt is made to generalize this result to more general quasilinear differential equations of the form (p(t)|x?|??1x?)? = q(t)|x|??1x.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

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.

scholarly journals On estimation of the variances for critical branching processes with immigration

2010 ◽  
Vol 47 (02) ◽  
pp. 526-542
Author(s):  
Chunhua Ma ◽  
Longmin Wang
Keyword(s):  
Least Squares ◽  
Branching Process ◽  
Branching Processes ◽  
Limiting Distributions ◽  
Least Squares Estimators ◽  
Conditional Least Squares ◽  
Regularly Varying ◽  
Branching Process With Immigration ◽  
Critical Branching Process

The conditional least-squares estimators of the variances are studied for a critical branching process with immigration that allows the offspring distributions to have infinite fourth moments. We derive different forms of limiting distributions for these estimators when the offspring distributions have regularly varying tails with index α. In particular, in the case in which 2 &lt; α &lt; 8/3, the normalizing factor of the estimator for the offspring variance is smaller than √n, which is different from that of Winnicki (1991).

Diffusion limits of critical branching processes conditioned on extinction in the near future

1976 ◽  
Vol 13 (02) ◽  
pp. 247-254
Author(s):  
Warren W. Esty
Keyword(s):  
Branching Processes ◽  
Limit Laws ◽  
Age Dependent ◽  
Diffusion Limits ◽  
Near Future

Limit laws and limiting diffusions are obtained for critical branching processes, either Galton-Watson or age-dependent, conditioned on extinction in the interval (T, cT], 1 &lt; c, as T→∞, and also as T→∞ and c ↓ 1.

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