Controlled Branching Processes

The limit behaviour of a controlled branching process with random control function is investigated. A necessary condition and a sufficient condition for the geometric growth of such a process are established by considering the L1-convergence. Finally, taking into account the classical X log+X criterion in branching processes, a necessary and sufficient condition is provided.

Branching processes in a random environment are natural generalisations of Galton-Watson processes. In this paper we analyse the asymptotic decay of the survival probability for a sequence of slightly supercritical branching processes in an iid random environment, where the offspring expectation converges from above to 1. We prove that Haldane’s asymptotics, known from classical Galton-Watson processes, turns up again in the random environment case, provided that one stays away from the critical/subcritical regime. A central building block is a connection to and a limit theorem for perpetuities with asymptotically vanishing interest rates.

Limiting distribution for subcritical controlled branching processes with random control function

On the Accuracy of Gaussian Approximation to the Distribution Functions of Sums of Independent Variables

Large deviation rates for supercritical and critical branching processes.- How fast does a general branching random walk spread?.- Boltzmann-Gibbs weights in the branching random walk.- Stochastic monotonicity and branching processes.- Multilevel multitype branching models of an information system.- On the shape of the wavefront of branching random walk.- Limiting distributions in branching processes with two types of particles.- Depth-first search of random trees, and Poisson point processes.- Towards dependence in general branching processes.- A criterion of boundedness of discrete branching random walk.- Quasistationarity in a branching model of division-within-division.- Population and density dependent branching processes.- Directed polymers in random media and spin glass models on trees..- A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes.- On two measures defined on the boundary of a branching tree.- Which critically branching populations persist?.- A simple path to Biggins' martingale convergence for branching random walk.- Unsolved problems concerning random walks on trees.- Branching processes with local dependencies.- Sharpness of second moment criteria for branching and tree-indexed processes.- On the recognition and structure of probability generating functions.- Record values of a family of branching processes.- Limit skeleton for critical crump-mode-Jagers branching processes.- Markov cascades.- Limit theorems for branching processes with random migration stopped at zero.

Previous article Next article On the Extension of the Class of Stable DistributionsV. M. KruglovV. M. Kruglovhttps://doi.org/10.1137/1117081PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] B. V. Gnedenko and , A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, Mass., 1962 Google Scholar[2] Yu. V. Linnik, Decomposition of probability distributions, Edited by S. J. Taylor, Dover Publications Inc., New York, 1964xii+242 MR0189085 (32:6512) 0121.35502 Google Scholar[3] B. V. Gnedenko, Some theorems on distribution functions, Učenie Zapiski Moskovskogo Univ., 30 (1939), 61–82, (In Russian.) Google Scholar[4] V. M. Kruglov, A note on infinitely divisible distributions, Theory Prob. Applications, 15 (1970), 319–324 10.1137/1115038 0329.60011 LinkGoogle Scholar[5] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– MR0322926 (48:1287) 0219.60027 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Functional L.I.L. for the Upper Extremes of i.i.d. Random Variarbles from a Sub-class of Heavy Tailed Distributions31 October 2021 | Journal of the Indian Society for Probability and Statistics, Vol. 22, No. 2 Cross Ref Functional law of the iterated logarithm for partial sum processes of non-negative valued iid random variables belonging to the domain of partial attraction of a semi-stable lawStatistics & Probability Letters, Vol. 137 Cross Ref A Log Log Law for Subsequences of Delayed Random Sums27 July 2017 | Journal of the Indian Society for Probability and Statistics, Vol. 18, No. 2 Cross Ref Merging to Semistable ProcessesI. Fazekas15 November 2012 | Theory of Probability & Its Applications, Vol. 56, No. 4AbstractPDF (209 KB)A Precise Asymptotic Behaviour of the Large Deviation Probabilities for Weighted SumsApplied Mathematics, Vol. 02, No. 09 Cross Ref A note on asymptotics of linear combinations of iid random variables18 March 2010 | Periodica Mathematica Hungarica, Vol. 60, No. 1 Cross Ref Merging of Linear Combinations to Semistable Laws3 January 2008 | Journal of Theoretical Probability, Vol. 22, No. 3 Cross Ref Merging asymptotic expansions for semistable random variables11 March 2009 | Lithuanian Mathematical Journal, Vol. 49, No. 1 Cross Ref Law of iterated logarithm and almost sure limit points for random sumsStatistical Methodology, Vol. 3, No. 4 Cross Ref On a functional equation generalizing the class of semistable distributionsAnnals of the Institute of Statistical Mathematics, Vol. 57, No. 4 Cross Ref Merging to Semistable LawsS. Csorgo and Z. 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The number Y n of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Y n as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Y n and EY n provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

Previous article Next article Second-Order Markov and Reciprocal Stationary Gaussian ProcessesR. N. MiroshinR. N. Miroshinhttps://doi.org/10.1137/1124099PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. N. Bernshtei˘n, On the dependencies among random variablesCollected Works, Vol. IV, Nauka, Moscow, 1964, 235–254, (In Russian.) Google Scholar[2] J. L. Doob, Stochastic processes, John Wiley & Sons Inc., New York, 1953viii+654 MR0058896 0053.26802 Google Scholar[3] Benton Jamison, Reciprocal processes: The stationary Gaussian case, Ann. Math. Statist., 41 (1970), 1624–1630 MR0267637 0248.60030 CrossrefGoogle Scholar[4] I. I. Gikhman and , A. V. Skorokhod, The Theory of Stochastic Processes, Vol. I, Springer-Verlag, New York, 1974 CrossrefGoogle Scholar[5] E. Kamke, Differentialgleichungen. Lösungsmethoden and Lösungen, Vol. 1, Chelsea, New York, 1974 Google Scholar[6] E. Wong, Some results concerning the zero-crossings of Gaussian noise, SIAM J. Appl. Math., 14 (1966), 1246–1254 10.1137/0114099 MR0207059 0147.15905 LinkGoogle Scholar[7] E. Wong, The distribution of intervals between zeros for a stationary Gaussian process, SIAM J. Appl. Math., 18 (1970), 67–73 10.1137/0118008 MR0256456 0221.60024 LinkGoogle Scholar[8] R. N. Miroshin, Convergence of Rice and Longuet-Higgins series for a Wong process, Theory Prob. Applications, 21 (1976), 863–866 10.1137/1121103 0374.60049 LinkGoogle Scholar[9] R. N. Miroshin, Asymptotics of the second factorial moment of the number of upward crossings of the line $kt+a$ by a Gaussian stationary process, Vestnik Leningrad. Univ., 23 (1968), 109–120 MR0235612 0165.52602 Google Scholar[10] M. Krein, Sur le problème du prolongement des fonctions hermitiennes positives et continues, C. R. (Doklady) Acad. Sci. URSS (N.S.), 26 (1940), 17–22 MR0004333 0022.35302 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Characterization of Multivariate Stationary Gaussian Reciprocal DiffusionsJournal of Multivariate Analysis, Vol. 62, No. 1 Cross Ref Gaussian reciprocal diffusions and positive definite sturm-liouville operators4 April 2007 | Stochastics and Stochastic Reports, Vol. 55, No. 3-4 Cross Ref Gaussian reciprocal processes revisitedStatistics & Probability Letters, Vol. 12, No. 4 Cross Ref Reciprocal covariance solutions of some matrix differential equationsStochastic Processes and their Applications, Vol. 37, No. 1 Cross Ref Characterization and Classification of Gaussian Second Order Reciprocal Processes Cross Ref Gaussian reciprocal processes and self-adjoint stochastic differential equations of second order2 May 2007 | Stochastics and Stochastic Reports, Vol. 34, No. 1-2 Cross Ref Reciprocal diffusions and stochastic differential equations of second order ∗Stochastics, Vol. 24, No. 4 Cross Ref Multivariate reciprocal stationary Gaussian processesJournal of Multivariate Analysis, Vol. 23, No. 1 Cross Ref On Miroshin’s Second-Order Reciprocal ProcessesJulia Abrahams3 August 2006 | SIAM Journal on Applied Mathematics, Vol. 44, No. 1AbstractPDF (316 KB)The Use of Rice SeriesR. N. Miroshin28 July 2006 | Theory of Probability & Its Applications, Vol. 28, No. 4AbstractPDF (1011 KB)The zero-crossing problem for some nonstationary Gaussian processes (Corresp.)IEEE Transactions on Information Theory, Vol. 28, No. 4 Cross Ref Convergence of Longuet-Higgins Series for Stationary Gaussian Markov Processes of First OrderR. N. Miroshin17 July 2006 | Theory of Probability & Its Applications, Vol. 26, No. 1AbstractPDF (1546 KB) Volume 24, Issue 4| 1980Theory of Probability & Its Applications History Submitted:11 July 1977Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1124099Article page range:pp. 845-852ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Previous article Next article Mathematical Aspects of the Reliability ProblemR. F. DrenickR. F. Drenickhttps://doi.org/10.1137/0108008PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. Henney, Reliability Factors for Ground Electronic Equipment, McGraw-Hill, New York, 1956 Google Scholar[2] G. E. Albert and , Lewis Nelson, Contributions to the statistical theory of counter data, Ann. Math. Statistics, 24 (1953), 9–22 MR0053447 0050.13905 CrossrefISIGoogle Scholar[3] R. F. Drenick, A statistical view of reliability, Proceedings of the Working Conference on the Theory of Reliability, New York Univ., 1957, 30–40, April Google Scholar[4] Willy Feller, On the integral equation of renewal theory, Ann. Math. Statistics, 12 (1941), 243–267 MR0005419 0026.23001 CrossrefGoogle Scholar[5] A. J. Lotka, A contribution to the theory of self-renewing aggregates, with special reference to industrial replacement, Ann. Math. Statist., 10 (1939), 1–25 0020.24604 CrossrefGoogle Scholar[6] E. B. Kurtz, Life Expectancy of Physical Equipment, Ronald Press, New York, 1930 Google Scholar[7] C. E. Weitz, Lamp Bulletin, Application Engineering Bulletin, Large Lamp Dept., General Electric Co., Cleveland 12, Ohio, 1956 Google Scholar[8] A. H. Benner and , B. E. Meredith, Designing reliability into electronic circuits, Proceedings of the National Electronics Conference, 1954, 137–145 Google Scholar[9] C. R. Knight, , E. R. Jervis and , G. R. Herd, The definition of terms of interest in the study of reliability, I.R.E. Trans. on Reliability and Quality Control, (1955), 34–56, April CrossrefGoogle Scholar[10] D. K. Gannett, Determination of the average life of vacuum tubes, Bell Laboratories Record, 18 (1940), 378–382 Google Scholar[11] Philip M. Morse, Queues, inventories and maintenance. The analysis of operational systems with variable demand and supply, Publications in Operations Research, Operations Research Society of America, No. 1, John Wiley & Sons Inc., New York, 1958ix+202 MR0091216 Google Scholar[12] George H. Weiss, On the theory of replacement of machinery with a random failure time, Naval Res. Logist. Quart., 3 (1956), 279–293 (1957) MR0084438 CrossrefGoogle Scholar[13] I. Richard Savage, Cycling, Naval Res. Logist. Quart., 3 (1956), 163–175 (1957) MR0086715 CrossrefGoogle Scholar[14] Richard Bellman, Dynamic programming, Princeton Univeristy Press, Princeton, N. J., 195710, 42 MR0090477 Google Scholar[15] William Feller, Fluctuation theory of recurrent events, Trans. Amer. Math. Soc., 67 (1949), 98–119 MR0032114 0039.13301 CrossrefISIGoogle Scholar[16] B. J. Flehinger, Reliability improvement through redundancy at various system levels, IBM J. Res. Develop., 2 (1958), 148–158 CrossrefISIGoogle Scholar[17] F. Moskowitz and , J. B. McLean, Some reliability aspects of systems design, I.R.E. Trans. on Reliability and Quality Control, (1956), 7–35, Sept. CrossrefGoogle Scholar[18] Conny Palm, Intensitätsschwankungen im Fernsprechverkehr, Ericsson Technics, 44 (1943), 189– MR0011402 0063.06088 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Analytic Properties of a Generating Function for the Number of Renewals17 July 2006 | Theory of Probability & Its Applications, Vol. 8, No. 1AbstractPDF (400 KB) Volume 8, Issue 1| 1960Journal of the Society for Industrial and Applied Mathematics History Submitted:07 October 1958Published online:10 July 2006 InformationCopyright © 1960 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0108008Article page range:pp. 125-149ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

We establish a general sufficient condition for a sequence of Galton–Watson branching processes in varying environments to converge weakly. This condition extends previ- ous results by allowing offspring distributions to have infinite variance. Our assumptions are stated in terms of pointwise convergence of a triplet of two real- valued functions and a measure. The limiting process is characterized by a backwards integro-differential equation satisfied by its Laplace exponent, which generalizes the branching equation satisfied by continuous state branching processes. Several examples are discussed, namely branching processes in random environment, Feller diffusion in varying environments and branching processes with catastrophes.

The development of a blood cell line originating from a pluripotent stem cell pool is modelled by a chain of multidimensional branching processes in which the sojourn times of the cells in certain resting states depend on the size of the following subpopulation. The stability of such a model is discussed qualitatively and some considerations concerning a possible malignant degeneration are presented. The behaviour of models for normal and malignant cell production are illustrated by stochastic stimulations.

A Developmentally Regulated Two-Step Process Generates a Noncentrosomal Microtubule Network in Drosophila Tracheal Cells

Mating functions considered by Asmussen (1980) and Daley (1968) for the class of bisexual Galton-Watson branching processes (GWBP) are shown to be superadditive. Consideration of a process that allows only sibling mating leads to a necessary condition for almost sure extinction in bisexual GWBP governed by superadditive mating functions. A simple example shows that the condition is not sufficient.

Previous article Next article Asymptotic Distributions of Curvatures of a Homogeneous Gaussian Random Field at Points of High Local MaximaV. P. NoskoV. P. Noskohttps://doi.org/10.1137/1129110PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Yu. Belyaev, Point processes and first passage problems, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, Univ. California Press, Berkeley, Calif., 1972, 1–17, Los Angeles 53:4207 Google Scholar[2] Robert J. Adler, The geometry of random fields, John Wiley & Sons Ltd., Chichester, 1981xi+280 82h:60103 0478.60059 Google Scholar[3] A. N. Mal'kov, An investigation of the curvature of Gaussian random processes and fields at local maximum pointsInvestigations on Random Fields, MGU Press, Moscow, 1974, 103–111, (In Russian.) Google Scholar[4] P. K. Rashevskii, A Course of Differential Geometry, GITTL, Moscow—Leningrad, 1950, (In Russian.) Google Scholar[5] Yu. K. Belyaev, Random point sets and problems of the level crossing typeExcursions of Random Fields, MGU Press, Moscow, 1972, 9–37, (In Russian.) Google Scholar[6] V. P. Nosko, Local structure of Gaussian homogeneous fields in the vicinity of high-level light sources, Soviet Math. Dokl, 10 (1969), 1481–1482 0267.60036 Google Scholar[7] V. P. Nosko, The characteristics of excursions of Gaussian homogeneous fields beyond a high level, Soviet-Japan Symposium on Probability Theory (Khabarovsk, August, 1969), SO Akad Nauk SSSR, Novosibirsk, 1969, 209–215, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails On the Shape of a High Excursion of a Gaussian Stationary Process24 March 2021 | Journal of Mathematical Sciences, Vol. 254, No. 4 Cross Ref Volume 29, Issue 4| 1985Theory of Probability & Its Applications History Submitted:09 September 1982Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1129110Article page range:pp. 817-823ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics