Abstract
Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields (Probab. Theory Relat. Fields 79(4):509–542, 1988) model. Fix n≥1 and β>0. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate β(n−k)/n, where k is the distance from the node to the root. Denote by Z n (t) the number of nodes with no descendants at time t and let T n =β −1 nln(n/ln4)+(ln2)/(2β). We prove that 2−n Z n (T n +nτ), τ∈ℝ, converges to the Gompertz curve exp(−(ln2) e −βτ). We also prove a central limit theorem for the martingale associated to Z n (t).
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Aldous, D., Shields, P.: A diffusion limit for a class of randomly-growing binary trees. Probab. Theory Relat. Fields 79(4), 509–542 (1988)
Baxter, M.A., Wynn, R.F., Jowitt, S.N., Wraith, J.E., Fairbairn, L.J., Ellington, I.: Study of telomere length reveals rapid aging of human marrow stromal cells following in vitro expansion. Stem Cells 22(5), 675–682 (2004)
Best, K., Pfaffelhuber, P.: The Aldous-Shields model revisited with applications to cellular ageing. Electron. Commun. Probab. 15, 475–488 (2010)
Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1999)
Coe, J.B., Mao, Y.: Gompertz mortality law and scaling behavior of the Penna model. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 72(5 Pt 1), 051925 (2005).
Dean, D.S., Majumdar, S.N.: Phase transition in a generalized Eden growth model on a tree. J. Stat. Phys. 124, 1351–1376 (2006)
Gompertz, B.: On the nature of the function expressive of the law of human mortality, and on a new method of determining the value of life contingencies. Philos. Trans. R. Soc. Lond. 115, 513–585 (1825)
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288. Springer, Berlin (1987)
Harley, C.B., Vaziri, H., Counter, C.M., Allsopp, R.C.: The telomere hypothesis of cellular aging. Exp. Gerontol. 27(4), 375–382 (1992)
Hayflick, L.: The limited in vitro lifetime of human diploid cell strains. Exp. Cell Res. 37, 614–636 (1965)
Laird, A.K.: Dynamics of relative growth. Growth 29(3), 249–263 (1965)
Laird, A.K.: Dynamics of growth in tumors and in normal organisms. In: National Cancer Institute Monograph, vol. 30, pp. 15–28 (1969)
Portugal, R.D., Land, M.G.P., Svaiter, B.F.: A computational model for telomere-dependent cell-replicative aging. Biosystems 91(1), 262–267 (2008)
Wallenstein, S., Brem, H.: Statistical analysis of wound-healing rates for pressure ulcers. Am. J. Surg. 188(1A Suppl), 73–78 (2004)
Winsor, C.P.: The Gompertz curve as a growth curve. Proc. Natl. Acad. Sci. USA 18(1), 1–8 (1932)
Wright, S.: Book review. J. Am. Stat. Assoc. 21, 494 (1926)
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B.F. Svaiter was partially supported by CNPq grants 302962/2011-5 and 474944/2010-7, FAPERJ grant E-26/102.940/2011 and by PRONEX-Optimization.
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Landim, C., Portugal, R.D. & Svaiter, B.F. A Markovian Growth Dynamics on Rooted Binary Trees Evolving According to the Gompertz Curve. J Stat Phys 148, 565–578 (2012). https://doi.org/10.1007/s10955-012-0549-z
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DOI: https://doi.org/10.1007/s10955-012-0549-z