Critical Bellman-Harris Branching Process Research Articles

Reduced critical Bellman–Harris branching processes for small populations

Abstract A critical Bellman-Harris branching process {Z(t),t ≥ 0} with finite variance of the offspring number is considered. Assuming that 0 < Z(t) ≤ φ(t), where either φ(t) = o(t) as t → ∞ or φ(t) = at,a>0, we study the structure of the process where Z(s,t) is the number of particles in the initial process at moment s which either survive up to moment t or have a positive number of descendants at this moment.

On Renewal Matrices Connected with Branching Processes with Tails of Distributions of Different Orders

We study irreducible renewal matrices generated by matrices whose rows are proportional to various distribution functions. Such matrices arise in studies of multi-dimensional critical Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching processes, unsolved problems are known that correspond to the case in which the tails of some of the above mentioned distribution functions are integrable, while the other distributions lack this property.We assume that the heaviest tails are regularly varying at the infinity with parameter −β ∈ [−1, 0) and asymptotically proportional, while the other tails are infinitesimal with respect to them. Under a series of additional conditions, we describe asymptotic properties of the first and second order increments for the renewal matrices.

Critical Bellman-Harris branching processes with long-living particles

A critical indecomposable two-type Bellman-Harris branching process is considered in which the life-length of the first-type particles has finite variance while the tail of the life-length distribution of the second-type particles is regularly varying at infinity with parameter β ∈ (0, 1]. It is shown that, contrary to the critical indecomposable Bellman-Harris branching processes with finite variances of the life-lengths of particles of both types, the probability of observing first-type particles at a distant moment t is infinitesimally less than the survival probability of the whole process. In addition, a Yaglom-type limit theorem is proved for the distribution of the number of the first-type particles at moment t given that the population contains particles of the first type at this moment.

Teoremas Límites en Procesos de Bellman–Harris con Segundos Momentos Finitos

En este artículo estudiamos diferentes teoremas límites en un proceso crítico de Bellman-Harris con un sólo tipo de partculas y con segundos momentos finitos. Los límites encontrados se hallaron con base en los siguientes dos procesos: “Procesos bajo la condición de no extinción” y “procesos bajo la condición de extinción en el futuro cercano”. En la observación de estos dos procesos hemos tenido en cuenta los dos diferentes casos: τi := dit y τi := di ± t, donde t es un punto de tiempo y di ∈ (0,∞) son constantes fijas para todo i = 1, . . . , k. Para el caso τi := dit, el lema de comparación 2.3 de Esty es útil para investigar el comportamiento asintótico de la función de generatriz conjunta F (s1, . . . , τk), para t → ∞; para el caso τi := t + di, no. Para éste último caso encontramos otro lema de comparación (lema 4.3), que es la base para demostrar teoremas límites si τi := t ± di.

Critical Bellman–Harris branching processes with infinite variance allowing state-dependent immigration

The branching processes with state-dependent immigration are considered as alternating regenerative processes. The main purpose is to demonstrate some new “regenerative” methods. Critical Bellman–Harris branching processes with state-dependent immigration are investigated and new limit theorems are obtained in the case of an infinite offspring variance and possibly infinite mean of the immigrants.

A Critical Emigration Branching Process

Abstract A Bellman–Harris process is defined with each cell subject to emigration in accord with a simple renewal distribution. The emigration process may be made to act similar to a critical Bellman–Harris branching process under simple conditions. Under these conditions, the emigration process is called a critical emigration process. Various limit theorems, analogous to those for critical Bellman–Harris processes, are given for this process. Proofs are largely by analogy with an appropriate critical Bellman–Harris process. A critical emigration–immigration Bellman–Harris process is also considered.

Decomposable Critical Bellman–Harris Branching Process with Two Particle Types. II

Decomposable Critical Bellman–Harris Branching Process with Two Particle Types. II

Decomposable Critical Bellman-Harris Branching Process with Two Types of Particles, I

Decomposable Critical Bellman-Harris Branching Process with Two Types of Particles, I

Asymptotic Behavior of a Class of Infinitely Divisible Distributions

Previous article Next article Asymptotic Behavior of a Class of Infinitely Divisible DistributionsA. L. YakymivA. L. Yakymivhttps://doi.org/10.1137/1132095PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. S. Gradshtein and , I. M. Ryzhik, Tables of Integrals, Sums, Series and Products, Fizmatgiz, Moscow, 1963, (In Russian.) Google Scholar[2] V. M. Zolotarev, On the asymptotic behavior of a class of infinitely divisible laws, Theory Probab. Appl., 6 (1961), 304–307 10.1137/1106039 0124.34501 LinkGoogle Scholar[3] E. Seneta, Regularly Varying Functions, Nauka, Moscow, 1985, (In Russian.) Google Scholar[4] William Feller, An introduction to probability theory and its applications. Vol. II. , Second edition, John Wiley & Sons Inc., New York, 1971xxiv+669 42:5292 0219.60003 Google Scholar[5] A. L. Yakymiv, Asymptotic behavior of the probability of continuing critical Bellman-Harris branching processes, Trudy Mat. Inst. Steklov, 177 (1986), 177–205, (In Russian.) 0607.60075 Google Scholar[6] A. L. Yakymiv, Asymptotic properties of the times the states change in a random record process, Theory Probab. Appl., 31 (1986), 508–512 10.1137/1131068 0623.60062 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Tail behaviors of semi-stable distributionsJournal of Mathematical Analysis and Applications, Vol. 393, No. 1 | 1 Sep 2012 Cross Ref On the Asymptotics of the Density of an Infinitely Divisible Distribution at InfinityA. L. YakymivTheory of Probability & Its Applications, Vol. 47, No. 1 | 25 July 2006AbstractPDF (122 KB)Remarks on the absolute maximum of a Lévy processJournal of Applied Probability, Vol. 39, No. 2 | 14 July 2016 Cross Ref A Probabilistic Approach to the Tails of Infinitely Divisible LawsSums, Trimmed Sums and Extremes | 1 Jan 1991 Cross Ref Volume 32, Issue 4| 1988Theory of Probability & Its Applications573-758 History Submitted:23 December 1985Published online:17 July 2006 InformationCopyright © 1987 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1132095Article page range:pp. 628-639ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

A Critical Bellman–Harris Branching Process with Particles of Final Type

Previous article Next article A Critical Bellman–Harris Branching Process with Particles of Final TypeV. A. VatutinV. A. Vatutinhttps://doi.org/10.1137/1131057PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] B. A. Sevast'yanov, Verzweigungsprozesse, Akademie-Verlag, Berlin, 1974xi+326 53:11785 Google Scholar[2] V. A. Vatutin, Limit theorems for a critical Bellman–Harris branching process with infinite variance, Theory Probab. Appl., 21 (1976), 839–842 10.1137/1121098 0383.60079 LinkGoogle Scholar[3] V. A. Vatutin, Discrete limit distributions of the number of particles in critical Bellman–Harris branching processes, Theory Probab. Appl., 22 (1977), 146–151 10.1137/1122014 0391.60082 LinkGoogle Scholar[4] V. A. Vatutin, A new limit theorem for a critical Bellman-Harris branching process, Mat. Sb. (N.S.), 109(151) (1979), 440–452, 480, (In Russian.) 81b:60085 Google Scholar[5] V. A. Vatutin and , S. M. Sagitov, A decomposable critical branching process with two types of particles, Trudy Mat. Inst. Steklov., 177 (1986), 3–20, 207, (In Russian.) 87k:60196 0619.60079 Google Scholar[6] A. G. Pakes, A limit theorem for the integral of a critical age-dependent branching process, Math. Biosci., 13 (1972), 109–112 10.1016/0025-5564(72)90026-0 48:7404 0228.60034 CrossrefGoogle Scholar[7] Krishna B. Athreya and , Peter E. Ney, Branching processes, Springer-Verlag, New York, 1972xi+287 51:9242 0259.60002 CrossrefGoogle Scholar[8] S. V. Nagaev, Transition phenomena for age-dependent branching processes with discrete time. I, Sibirsk. Mat. Ž., 15 (1974), 368–394, 461, (In Russian.) 49:4108 Google Scholar[9] E. Seneta, Regularly Varying Functions, Nauka, Moscow, 1972, (In Russian.) Google Scholar[10] V. V. Petrov, Sums of independent random variables, Springer-Verlag, New York, 1975x+346 52:9335 0322.60042 CrossrefGoogle Scholar[11] A. L. Yakymiv, The asymptotic behavior of the probability of continuation of critical Bellman-Harris branching processes, Trudy Steklov Mat. Inst., 177 (1986), 177–205, (In Russian.) 0607.60075 Google Scholar[12] William Feller, An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons Inc., New York, 1971xxiv+669 42:5292 0219.60003 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Branching Processes with Final Types of Particles and Random Trees17 July 2006 | Theory of Probability & Its Applications, Vol. 39, No. 4AbstractPDF (1118 KB)The Total Number of Particles in a Reduced Bellman-Harris Branching Process17 July 2006 | Theory of Probability & Its Applications, Vol. 38, No. 3AbstractPDF (407 KB) Volume 31, Issue 3| 1987Theory of Probability & Its Applications History Submitted:23 December 1985Published online:28 July 2006 InformationCopyright © 1987 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1131057Article page range:pp. 428-438ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Asymptotic Properties of the Times the States Change in a Random Record Process

Previous article Next article Asymptotic Properties of the Times the States Change in a Random Record ProcessA. L. YakymivA. L. Yakymivhttps://doi.org/10.1137/1131068PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Donald P. Gaver, Random record models, J. Appl. Probability, 13 (1976), 538–547 54:8841 0399.60083 CrossrefGoogle Scholar[2] P. Embrechts and , E. Omey, On subordinated distributions and random record processes, Math. Proc. Cambridge Philos. Soc., 93 (1983), 339–353 84h:60042 0513.60029 CrossrefGoogle Scholar[3] M. Westcott, The random record model, Proc. Roy. Soc. London Ser. A, 356 (1977), 529–547 58:7837 0369.60062 CrossrefGoogle Scholar[4] Mark Westcott, On the tail behaviour of record-time distributions in a random record process, Ann. Probab., 7 (1979), 868–873 80j:60037 0416.60061 CrossrefGoogle Scholar[5] Eugene Seneta, Regularly varying functions, Springer-Verlag, Berlin, 1976v+112, Lecture Notes in Mathematics 508, Heidelberg–New York 56:12189 0324.26002 CrossrefGoogle Scholar[6] A. L. Yakymiv, Asymptotic behavior of the probability of continuation of critical Bellman–Harris branching processes, Trudy Mat. Inst. Steklov, 177 (1986), 177–205, (In Russian.) 0607.60075 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Bibliography14 October 2011 Cross Ref On the convergence of positive increasing functions to infinity20 April 2011 | Ukrainian Mathematical Journal, Vol. 62, No. 10 Cross Ref PRV property and the ϕ-asymptotic behavior of solutions of stochastic differential equationsLithuanian Mathematical Journal, Vol. 47, No. 4 Cross Ref On the Asymptotics of the Density of an Infinitely Divisible Distribution at InfinityA. L. Yakymiv25 July 2006 | Theory of Probability & Its Applications, Vol. 47, No. 1AbstractPDF (122 KB)Record sequences and their applications Cross Ref Asymptotics of the kth Record TimesA. L. Yakymiv12 July 2006 | Theory of Probability & Its Applications, Vol. 40, No. 4AbstractPDF (298 KB)Asymptotic Behavior of a Class of Infinitely Divisible DistributionsA. L. Yakymiv17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 4AbstractPDF (680 KB)RecordsV. B. Nevzorov17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 2AbstractPDF (3250 KB) Volume 31, Issue 3| 1987Theory of Probability & Its Applications History Submitted:08 April 1985Published online:28 July 2006 InformationCopyright © 1987 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1131068Article page range:pp. 508-512ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Reduced Critical Bellman–Harris Branching Process with Several Types of Particles

Reduced Critical Bellman–Harris Branching Process with Several Types of Particles

Critical Bellman-Harris branching processes starting with a large number of particles

Critical Bellman-Harris branching processes starting with a large number of particles

Two limit theorems for critical Bellman-Harris branching processes

Two limit theorems for critical Bellman-Harris branching processes

A Class of Limit Theorems for a Critical Bellman–Harris Branching Process

Previous article Next article A Class of Limit Theorems for a Critical Bellman–Harris Branching ProcessV. A. VatutinV. A. Vatutinhttps://doi.org/10.1137/1126087PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. A. Vatutin, Limit theorems for a critical Bellman-Harris branching process with infinite variance, Theory Prob. Appl., 21 (1976), 839–842 0383.60079 LinkGoogle Scholar[2] V. A. Vatutin, Discrete limit distributions of the particle number in critical Bellman-Harris branching processes, Theory Prob. Appl., 22 (1977), 146–152 0391.60082 LinkGoogle Scholar[3] V. A. Vatutin, A new limit theorem for a critical Bellman-Harris branching process, Mat. Sb. (N.S.), 109(151) (1979), 440–452, 480 81b:60085 0443.60080 Google Scholar[4] S. V. Nagaev, Transition phenomena for age-dependent branching processes with discrete time. II, Sibirsk. Mat. Ž., 15 (1974), 570–579, 701 50:14975 0294.60065 Google Scholar[5] W. Feller, An Introduction to the Theory of Probability and Its Applications, Vol. II, John Wiley, New York, 1966 Google Scholar[6] T. Harris, The theory of branching processes, Die Grundlehren der Mathematischen Wissenschaften, Bd. 119, Springer-Verlag, Berlin, 1963xiv+230 29:664 0117.13002 CrossrefGoogle Scholar[7] B. V. Shabat, Introduction to Complex Analysis, Part 1, Nauka, Moscow, 1976, (In Russian.) Google Scholar[8] Martin I. Goldstein, Critical age-dependent branching processes: Single and multitype, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 17 (1971), 74–88 43:4132 0212.19704 CrossrefGoogle Scholar[9] Martin I. Goldstein and , Fred M. Hoppe, Limit theorems for the critical age-dependent branching process with infinite variance, Stochastic Processes Appl., 5 (1977), 297–305 10.1016/0304-4149(77)90037-0 56:16811 0369.60101 CrossrefGoogle Scholar[10] Eugene Seneta, Regularly varying functions, Springer-Verlag, Berlin, 1976, 88– 56:12189 CrossrefGoogle Scholar[11] R. S. Slack, A branching process with mean one and possibly infinite variance, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 9 (1968), 139–145 37:3661 0164.47002 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Decomposable Critical Bellman–Harris Branching Process with Two Particle Types. IIV. A. Vatutin and S. M. SagitovTheory of Probability & Its Applications, Vol. 34, No. 2 | 17 July 2006AbstractPDF (790 KB) Volume 26, Issue 4| 1982Theory of Probability & Its Applications657-870 History Submitted:27 March 1980Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1126087Article page range:pp. 806-812ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Reduced Branching Processes

Previous article Next article Reduced Branching ProcessesA. L. YakymivA. L. Yakymivhttps://doi.org/10.1137/1125069PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons Inc., New York, 1968xii+253 38:1718 0172.21201 Google Scholar[2] V. A. Vatutin, Limit theorems for critical multitype Markov branching processes with infinite second moments, Mat. Sb. (N.S.), 103(145) (1977), 253–264, 319, (In Russian.) 56:1488 Google Scholar[3] B. A. Sewast'yanow, Verzweigungsprozesse, Akademie-Verlag, Berlin, 1974xi+326, (German translation), Branching Processes, Nauka, Moscow, 1971. (In Russian.) 53:11785 Google Scholar[4] Klaus Fleischmann and , Rainer Siegmund-Schultze, The structure of reduced critical Galton-Watson processes, Math. Nachr., 79 (1977), 233–241 57:1674 0299.60065 CrossrefGoogle Scholar[5] R. R. Rao, Relations between weak and uniform convergence of measures with applications, Ann. Math. Statist., 33 (1962), 659–680 25:1258 CrossrefGoogle Scholar[6] Eugene Seneta, Regularly varying functions, Springer-Verlag, Berlin, 1976v+112, Lecture Notes in Mathematics, 508 56:12189 0324.26002 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Typical Behavior of the Harmonic Measure in Critical Galton–Watson Trees with Infinite Variance Offspring Distribution16 March 2017 | Journal of Theoretical Probability, Vol. 31, No. 3 Cross Ref The Structure of the Decomposable Reduced Branching Processes. I. Finite-Dimensional Distributions19 November 2015 | Theory of Probability & Its Applications, Vol. 59, No. 4AbstractPDF (254 KB)The harmonic measure of balls in critical Galton-Watson trees with infinite variance offspring distributionElectronic Journal of Probability, Vol. 19, No. none Cross Ref Reduced Branching Processes with Very Heavy Tails14 July 2016 | Journal of Applied Probability, Vol. 45, No. 1 Cross Ref Reduced Branching Processes in Random Environment: The Critical Case25 July 2006 | Theory of Probability & Its Applications, Vol. 47, No. 1AbstractPDF (180 KB)The general coalescent with asynchronous mergers of ancestral lines14 July 2016 | Journal of Applied Probability, Vol. 36, No. 4 Cross Ref The Total Number of Particles in a Reduced Bellman-Harris Branching Process17 July 2006 | Theory of Probability & Its Applications, Vol. 38, No. 3AbstractPDF (407 KB)Reduced Critical Bellman–Harris Branching Process with Several Types of Particles28 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 4AbstractPDF (868 KB)Asymptotic Properties of Subcritical and Supercritical Reduced Branching Processes28 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 1AbstractPDF (572 KB) Volume 25, Issue 3| 1981Theory of Probability & Its Applications History Submitted:11 May 1979Published online:28 July 2006 InformationCopyright © 1981 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1125069Article page range:pp. 584-588ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On a Class of Critical Bellman–Harris Branching Processes with Several Types of Particles

On a Class of Critical Bellman–Harris Branching Processes with Several Types of Particles

A limit Theorem for a Critical Bellman-Harris Branching Process with Several Types of Particles and Infinite Second Moments

A limit Theorem for a Critical Bellman-Harris Branching Process with Several Types of Particles and Infinite Second Moments

Discrete Limit Distributions of the Number of Particles in Critical Bellman–Harris Branching Process

Discrete Limit Distributions of the Number of Particles in Critical Bellman–Harris Branching Process

Limit Theorems for a Critical Bellman-Harris Branching Process with Infinite Variance

Limit Theorems for a Critical Bellman-Harris Branching Process with Infinite Variance