The Annals of Applied Probability

Relative frequencies in multitype branching processes

Andrei Y. Yakovlev and Nikolay M. Yanev

Full-text: Open access

Abstract

This paper considers the relative frequencies of distinct types of individuals in multitype branching processes. We prove that the frequencies are asymptotically multivariate normal when the initial number of ancestors is large and the time of observation is fixed. The result is valid for any branching process with a finite number of types; the only assumption required is that of independent individual evolutions. The problem under consideration is motivated by applications in the area of cell biology. Specifically, the reported limiting results are of advantage in cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement. Relevant statistical applications are discussed in the context of asymptotic maximum likelihood inference for multitype branching processes.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 1 (2009), 1-14.

Dates
First available in Project Euclid: 20 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1235140330

Digital Object Identifier
doi:10.1214/08-AAP539

Mathematical Reviews number (MathSciNet)
MR2498669

Zentralblatt MATH identifier
1159.60345

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx]
Secondary: 62P10: Applications to biology and medical sciences 92D25: Population dynamics (general)

Keywords
Multitype branching processes relative frequencies joint asymptotic normality cell proliferation

Citation

Yakovlev, Andrei Y.; Yanev, Nikolay M. Relative frequencies in multitype branching processes. Ann. Appl. Probab. 19 (2009), no. 1, 1--14. doi:10.1214/08-AAP539. https://projecteuclid.org/euclid.aoap/1235140330


Export citation

References

  • [1] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • [2] Borovkov, A. A. (1998). Probability Theory. Gordon and Breach Science Publishers, Amsterdam. Translated from the 1986 Russian original by O. Borovkova and revised by the author.
  • [3] Crump, K. S. and Mode, C. J. (1969). An age-dependent branching process with correlations among sister cells. J. Appl. Probab. 6 205–210.
  • [4] Dion, J.-P. and Yanev, N. M. (1994). Statistical inference for branching processes with an increasing random number of ancestors. J. Statist. Plann. Inference 39 329–351.
  • [5] Dion, J.-P. and Yanev, N. M. (1995). Central limit theorem for martingales in BGWR branching processes with some statistical applications. Math. Methods Statist. 4 344–358.
  • [6] Dion, J. P. and Yanev, N. M. (1997). Limit theorems and estimation theory for branching processes with an increasing random number of ancestors. J. Appl. Probab. 34 309–327.
  • [7] Feller, W. (1951). Diffusion processes in genetics. In Proc. Second Berkeley Sympos. Math. Statist. Probab. 1950 227–246. Univ. California Press, Berkeley.
  • [8] Feller, W. (1971). An Introduction to Probability and Its Applications 2, 2nd ed. Wiley, New York.
  • [9] Guttorp, P. (1991). Statistical Inference for Branching Processes. Wiley, New York.
  • [10] Haccou, P., Jagers, P. and Vatutin, V. (2005). Branching Processes: Variation, Growth and Extinction of Populations. Cambridge Univ. Press., Cambridge.
  • [11] Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.
  • [12] Hyrien, O., Mayer-Pröschel, M., Noble, M. and Yakovlev, A. (2005). Estimating the life-span of oligodendrocytes from clonal data on their development in cell culture. Math. Biosci. 193 255–274.
  • [13] Hyrien, O., Mayer-Pröschel, M., Noble, M. and Yakovlev, A. (2005). A stochastic model to analyze clonal data on multi-type cell populations. Biometrics 61 199–207.
  • [14] Hyrien, O., Ambescovich, I., Mayer-Proschel, M., Noble, M. and Yakovlev, I. (2006). Stochastic modeling of oligodendrocyte generation in cell culture: Model validation with time-lapse data. Theoret. Biol. Med. Model. 3.
  • [15] Jagers, P. (1969). The proportions of individuals of different kinds in two-type populations. A branching process problem arising in biology. J. Appl. Probab. 6 249–260.
  • [16] Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, London.
  • [17] Kimmel, M. and Axelrod, D. E. (2002). Branching Processes in Biology. Interdisciplinary Applied Mathematics 19. Springer, New York.
  • [18] Lamperti, J. (1967). Limiting distributions for branching processes. In Proc. Fifth Berkeley Sympos. Math. Statist. Probab. (Berkeley, Calif., 1965/66) II: Contributions to Probability Theory Part 2 225–241. Univ. California Press, Berkeley.
  • [19] Mode, C. J. (1971). Multitype Branching Processes. Theory and Applications. American Elsevier Publishing Co., Inc., New York.
  • [20] Mode, C. J. (1971). Multitype age-dependent branching processes and cell cycle analysis. Math. Biosci. 10 177–190.
  • [21] Quine, M. P. (1970). A note of the moment structure of the multitype Galton–Watson process. Biometrika 57 219–222.
  • [22] Yakovlev, A. Y. and Yanev, N. M. (1989). Transient Processes in Cell Proliferation Kinetics. Lecture Notes in Biomathematics 82. Springer, Berlin.
  • [23] Yakovlev, A. Y., Mayer-Proschel, M. and Noble, M. (1998). A stochastic model of brain cell differentiation in tissue culture. J. Math. Biol. 37 49–60.
  • [24] Yakovlev, A. Y., Boucher, K., Mayer-Proschel, M. and Noble, M. (1998). Quantitative insight into proliferation and differentiation of oligodendrocyte type 2 astrocyte progenitor cells in vitro. Proc. Natl. Acad. Sci. USA 95 144–167.
  • [25] Yakovlev, A. and Yanev, N. (2006). Branching stochastic processes with immigration in analysis of renewing cell populations. Math. Biosci. 203 37–63.
  • [26] Yanev, N. M. (1975). The statistics of branching processes. Teor. Verojatnost. i Primenen. 20 623–633.
  • [27] Yanev, N., Jordan, C. T., Catlin, S. and Yakovlev, A. (2005). Two-type Markov branching processes with immigration as a model of leukemia cell kinetics. C. R. Acad. Bulgare Sci. 58 1025–1032.
  • [28] Yanev, N. M. (2008). Statistical inference for branching processes. In Records and Branching Processes (M. Ahsanullah and G. P. Yanev, eds.). NOVA Science Publishers, Hauppauge, NY.