Branching Processes as Sums of Dependent Random Variables

1. The main notions.- 2. The main lemmas.- 2.1. General lemmas on the approximation of distribution of an arbitrary random variable by the normal distribution.- 2.2. Proof of lemmas 2.1-2.4.- 3. Theorems on large deviations for the distributions of sums of independent random variables.- 3.1. Theorems on large deviations under Bernstein's condition.- a) Sums of non-identically distributed random variables.- b) Sums of weighted random variables.- 3.2. A theorem of large deviations in terms of Lyapunov's fractions.- 4. Theorems of large deviations for sums of dependent random variables.- 4.1. Estimates of the kth order centered moments of random processes with mixing.- 4.2. Estimates of mixed cumulants of random processes with mixing.- 4.3. Estimates of cumulants of sums of dependent random variables.- 4.4. Theorems and inequalities of large deviations for sums of dependent random variables.- 5. Theorems of large deviations for polynomial forms, multiple stochastic integrals and statistical estimates.- 5.1. Estimates of cumulants and theorems of large deviations for polynomial forms, polynomial Pitman estimates and U-statistics.- 5.2. Cumulants of multiple stochastic integrals and theorems of large deviations.- 5.3. Large deviations for estimates of the spectrum of a stationary sequence.- 6. Asymptotic expansions in the zones of large deviations.- 6.1. Asymptotic expansion for distribution density of an arbitrary random variable.- 6.2. Estimates for characteristic functions.- 6.3. Asymptotic expansion in the Cramer zone for distribution density of sums of independent random variables.- 6.4. Asymptotic expansions in integral theorems with large deviations.- 7. Probabilities of large deviations for random vectors.- 7.1. General lemmas on large deviations for a random vector with regular behaviour of cumulants.- 7.2. Theorems on large deviations for sums of random vectors and quadratic forms.- a) Sums of non-identically distributed random vectors.- b) Sums of weighted random vectors.- c) Sums of random number of random vectors.- d) Quadratic forms.- Appendices.- Appendix 1. Proof of inequalities for moments and Lyapunov's fractions.- Appendix 2. Proof of the lemma on the representation of cumulants.- Appendix 3. Leonov - Shiryaev's formula.- References.

An approach is developed to construction of random point distribution in 3-dimensional space based on the theory of branching processes. Correlation functions of all orders have been obtained in general form. The neutron and Levy branching processes are numerically investigated. It is shown that fractal properties of the obtained distributions develop not only in the case of critical cascades but also in the case of subcritical, close to critical ones. In the last case, fractal properties appear in the limit region of distances. The difference between distributions generated by branching and nonbranching processes is discussed.

Quantitative Polymerase Chain Reaction (Q-PCR) aims at determining the initial quantity of specific nucleic acids from the observation of the number of amplified DNA molecules. The most widely used technology to monitor the number of DNA molecules as they replicate is based on fluorescence chemistry. Considering this measurement technique, the observation of DNA amplification by PCR contains intrinsically two kinds of variability. On the one hand, the number of replicated DNA molecules is random, and on the other hand, the measurement of the fluorescence emitted by the DNA molecules is collected with some random error. Relying on a stochastic model of these two types of variability, we aim at providing estimators of the parameters arising in the proposed model, and, more specifically, of the initial amount of molecules. The theory of branching processes is classically used to model the evolution of the number of DNA molecules at each replication cycle. The model is a binary splitting Galton-Watson branching process. Its unknown parameters are the initial number of DNA molecules and the reaction efficiency of PCR, which is defined as the probability of replication of a DNA molecule. The number of DNA molecules is indirectly observed through noisy fluorescence measurements resulting in a so-called Hidden Markov Model. We aim at inference of the parameters of the underlying branching process, and the parameters of the noise from the fluorescence measurements in a Bayesian framework. Using simulations and experimental data, we investigate the performance of the Bayesian estimators obtained by Markov Chain Monte Carlo methods.

This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in independent and identically distributed random environments. This is a natural generalization of the time-inhomogeneous branching processes. The key assumptions of the family of population models in question are nonoverlapping generations and discrete time. The reader should be aware of the fact that there are many very interesting papers covering other issues in the theory of branching processes in random environments which are not mentioned here.

Previous article Next article On a Condition of Almost Markov RegularityV. A. StatulyavichusV. A. Statulyavichushttps://doi.org/10.1137/1128030PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] B. Ryauba, The central limit theorem for sums of series of weakly dependent random variables, Litovsk. Mat. Sb., 2 (1962), 193–205, (In Russian.) 29:6526 Google Scholar[2] I. G. Zhurbenko, On the estimation of mixed cumulants for a class of random processes, Theory Prob. Appl., 15 (1970), 525–528 LinkGoogle Scholar[3] I. G. Zhurbenko, On strong estimates of mixed cumulants of random processes, Siberian Math. J., 13 (1972), 302–303 CrossrefGoogle Scholar[4] I. G. Zhurbenko and , N. M. Zuev, The higher spectral densities of stationary processes with mixing, Ukrain. Mat. Ž., 27 (1975), 452–463, 572 53:14620 0325.60039 Google Scholar[5] I. G. Zhurbenko, Mixing conditions and asymptotic theory of stationary time seriesStochastic processes and related topics (Proc. Summer Res. Inst. on Statist. Inference for Stochastic Processes, Indiana Univ., Bloomington, Ind., 1974, Vol. 1; dedicated to Jerzy Neyman), Academic Press, New York, 1975, 259–265 51:9177 0349.60031 Google Scholar[6] V. A. Statulyavichus, Limit theorems for random functions. I, Litovsk. Mat. Sb., 10 (1970), 583–592, (In Russian.) 43:2765 Google Scholar[7] V. A. Statulyavichus, Limit theorems for dependent random variables under various regularity conditions, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, 173–181 55:1420 Google Scholar[8] V. A. Statulyavichus, Application of semi-invariants to asymptotic analysis of distributions of random processesMultivariate analysis, IV (Proc. Fourth Internat. Sympos., Dayton, Ohio, 1975), North-Holland, Amsterdam, 1977, 325–337 57:1614 Google Scholar[9] R. L. Dobrushin, The description of a random field by means of conditional probabilities and conditions for its regularity, Theory Prob. Appl., 13 (1968), 197–224 LinkGoogle Scholar[10] I. A. Ibragimov, Some limit theorems for stationary processes, Theory Prob. Appl., 7 (1962), 349–382 LinkGoogle Scholar[11] M. Rosenblatt, Some remarks on a mixing condition, Ann. Probab., 7 (1979), 170–172 80k:60071a 0925.60052 CrossrefGoogle Scholar[12] V. A. Statulyavichus, Large-deviation theorems for sums of dependent random variables, Lithuanian Math. J., 19 (1979), 289–295 CrossrefGoogle Scholar[13] A. V. Bulinskii, Limit Theorems for Random Processes and Fields, MGU, Moscow, 1981, (In Russian.) Google Scholar[14] V. Statulyavichus, On limit theorems for dependent random variables, Abstracts of Communications of the Third International Vilnius Conference on Probability Theory and Mathematical Statistics, Vilnius: In-t matem. i kibern. AN Lit SSR, 1981, 320–323 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails An Almost-Markov-Type Mixing Condition and Large Deviations for Boolean Models on the LineActa Applicandae Mathematicae, Vol. 96, No. 1-3 | 23 March 2007 Cross Ref Approximation of Distributions of Sums of Weakly Dependent Random Variables by the Normal DistributionLimit Theorems of Probability Theory | 1 Jan 2000 Cross Ref Limit Theorems on Large DeviationsLimit Theorems of Probability Theory | 1 Jan 2000 Cross Ref Every “lower psi-mixing” Markov chain is “interlaced rho-mixing”Stochastic Processes and their Applications, Vol. 72, No. 2 | 1 Dec 1997 Cross Ref Random fields: Applications in cell biologyStochastic Spatial Processes | 16 September 2006 Cross Ref Volume 28, Issue 2| 1984Theory of Probability & Its Applications219-469 History Submitted:10 January 1983Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1128030Article page range:pp. 379-383ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables

The capacity of pluripotent embryonic stem cells to differentiate into any cell type in the body makes them invaluable in the field of regenerative medicine. However, because of the complexity of both the core pluripotency network and the process of cell fate computation it is not yet possible to control the fate of stem cells. We present a theoretical model of stem cell fate computation that is based on Halley and Winkler’s Branching Process Theory (BPT) and on Greaves et al.’s agent-based computer simulation derived from that theoretical model. BPT abstracts the complex production and action of a Transcription Factor (TF) into a single critical branching process that may dissipate, maintain, or become supercritical. Here we take the single TF model and extend it to multiple interacting TFs, and build an agent-based simulation of multiple TFs to investigate the dynamics of such coupled systems. We have developed the simulation and the theoretical model together, in an iterative manner, with the aim of obtaining a deeper understanding of stem cell fate computation, in order to influence experimental efforts, which may in turn influence the outcome of cellular differentiation. The model used is an example of self-organization and could be more widely applicable to the modelling of other complex systems. The simulation based on this model, though currently limited in scope in terms of the biology it represents, supports the utility of the Halley and Winkler branching process model in describing the behaviour of stem cell gene regulatory networks. Our simulation demonstrates three key features: (i) the existence of a critical value of the branching process parameter, dependent on the details of the cistrome in question; (ii) the ability of an active cistrome to “ignite” an otherwise fully dissipated cistrome, and drive it to criticality; (iii) how coupling cistromes together can reduce their critical branching parameter values needed to drive them to criticality.

A number of simple formulae concerning the growth rate, doubling time, and age distribution of cell populations are in common usage. Many of these are invalid unless the generation times have no variation, a situation which occurs rarely. The correct formulae, which are available from the theory of age-dependent branching processes, are often ignored in the biological literature, perhaps due to their complexity. In this paper the major formulae from the theory of age-dependent branching processes are reorganised in terms of the mean, coefficient of variation, and skewness of the generation times. This leads to a greater insight into the theory and some simple formulae which account for the variability.

The search for a result such as the one presented in this note was motivated by an application in the theory of Markov branching processes. The limiting behaviour of a Markov branching process is determined mainly by properties of the set of its first moments, usually given as a semigroup of non-negative, linear-bounded operators on a Banach space. The principal case is that in which these operators are in some sense primitive. If the underlying space is finite-dimensional, the case of primitivity is described to complete satisfaction by Perron's theorem. For more general spaces we have the well known extension of Perron's result by Kreĭn and Rutman (8). Unfortunately, the use of this extension in the theory of general branching processes has so far not led to limit theorems as strong as the best results known in the finite-dimensional case. At least for this specific purpose the classical Kreĭn-Rutman theorem seems to be too crude. In fact, already the simplest branching diffusions on bounded domains (3) suggest a more refined, though necessarily less general extension of Perron's theorem.

Previous article Next article The Central Limit Theorem for a Sequence of Random Variables with a Slowly Growing Number of DependencesM. B. Petrovskaya and A. M. LeontovichM. B. Petrovskaya and A. M. Leontovichhttps://doi.org/10.1137/1127089PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. B. Ivanov, Cell Foundations of Plant Growth, Nauka, Moscow, 1974, (In Russian.) Google Scholar[2] Z. A. Lomnicki and , S. K. Zaremba, A further instance of the central limit theorem for dependent random variables, Math. Z, 66 (1957), 490–494 20:6152 0077.33101 CrossrefGoogle Scholar[3] A. F. Ronzhin, On limit theorems for sums of dependent random variables, Theory Prob. Appl., 24 (1979), 549–559 0435.60023 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails When Janson meets McDiarmid: Bounded difference inequalities under graph-dependenceStatistics & Probability Letters, Vol. 17 Cross Ref Weighted dependency graphsElectronic Journal of Probability, Vol. 23, No. none Cross Ref Dependency graphs and mod-Gaussian convergence7 December 2016 Cross Ref A law of large numbers for nearest neighbour statistics12 August 2008 | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 464, No. 2100 Cross Ref Central limit theorems for k-nearest neighbour distancesStochastic Processes and their Applications, Vol. 85, No. 2 Cross Ref On the variance of the random sphere of influence graphRandom Structures and Algorithms, Vol. 14, No. 2 Cross Ref On a Theorem of JansonV. G. Mikhailov17 July 2006 | Theory of Probability & Its Applications, Vol. 36, No. 1AbstractPDF (378 KB)Asymptotic normality of some Graph-Related statistics14 July 2016 | Journal of Applied Probability, Vol. 26, No. 1 Cross Ref On a Condition for Slowly Growing Number of DependenciesA. M. Leontovich28 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 1AbstractPDF (623 KB) Volume 27, Issue 4| 1983Theory of Probability & Its Applications History Submitted:20 December 1977Published online:17 July 2006 InformationCopyright © 1983 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1127089Article page range:pp. 815-825ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

A necessary condition is given for the convergence of distributions of the sums of a random number of independent random variables. This is made on the basis of a theorem which gives sufficient conditions for the convergence of distributions of randomly stopped stochastic processes. The random indices are supposed to be independent of the sequence of summands.

Sufficient conditions are found for the applicability of the generalized law of the iterated logarithm for sums of dependent random variables in the case where the sequence of normalizing constants is not necessarily nondecreasing.

This paper investigates the joint tail behavior of infinite sums of random variables with random weights (infinite sums, for short) and random sums of random variables with random weights (random sums, for short), where the primary random variables are non-negative, heavy-tailed, and dependent. Under some appropriate conditions on non-negative random weights and the framework of a newly proposed dependence structure, we obtain some asymptotic formulas for the joint tail behavior of infinite sums and random sums. The obtained results are applied to derive an asymptotic formula, which unifies both finite-time and infinite-time cases of the ruin probability in a bi-dimensional risk model.

Capital $"O"$ and lower-case $"o"$ approximations of the expected value of a class of smooth functions $(f\in C^r(R))$ of the normalized random partial sums of dependent random variables by the expectation of the corresponding functions of Gaussian random variables are established. The same types of approximation are also obtained for dependent random vectors. This generalizes and improves previous results of the author (1980) and Rychlik and Szynal (1979).

In (Ordonez Cabrera and Volodin, J. Math. Anal. Appl. 305:644–658, 2005), the authors introduce the notion of h-integrability of an array of random variables with respect to an array of constants, and obtained some mean convergence theorems for weighted sums of random variables subject to some special kinds of dependence.