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Convergence results on multitype, multivariate branching random walks

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

J. D. Biggins  and
A. Rahimzadeh Sani
J. D. Biggins*
Affiliation:
The University of Sheffield
A. Rahimzadeh Sani*
Affiliation:
Teacher Training University of Tehran
*
∗ Postal address: Department of Probability and Statistics, The University of Sheffield, Sheffield S3 7RH, UK. Email address: j.biggins@sheffield.ac.uk
∗∗ Postal address: Department of Mathematics, Teacher Training University of Tehran, 49 Mofatteh Avenue, Tehran, 15614, Iran. Email address: rahimsan@saba.tmu.ac.ir
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Abstract

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We consider a multi-type branching random walk on d-dimensional Euclidian space. The~uniform convergence, as n goes to infinity, of a scaled version of the Laplace transform of the point process given by the nth generation particles of each type is obtained. Similar results in the one-type case, where the transform gives a martingale, were obtained in Biggins (1992) and Barral (2001). This uniform convergence of transforms is then used to obtain limit results for numbers in the underlying point processes. Supporting results, which are of interest in their own right, are obtained on (i) ‘Perron-Frobenius theory’ for matrices that are smooth functions of a variable λ∈L and are nonnegative when λ∈L−⊂L, where L is an open set in ℂd, and (ii) saddlepoint approximations of multivariate distributions. The saddlepoint approximations developed are strong enough to give a refined large deviation theorem of Chaganty and Sethuraman (1993) as a by-product.

Keywords

Large deviationlocal limit theoremsaddlepoint approximationuniform convergencemulti-type branching random walkLaplace transformmartingalePerron-Frobeniusnonnegative matrix

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 681 - 705
Copyright
Copyright © Applied Probability Trust 2005 

References

Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.Google Scholar
Barral, J. (2001). Generalized vector multiplicative cascades. Adv. Appl. Prob. 33, 874–895.Google Scholar
Bhattacharya, R. N. (1972). Recent results on refinements of the central limit theorem. In Proc. Sixth Berkeley Symp. Math. Statist. Prob. (Berkeley, CA, 1970/71), Vol. 2, University of California Press, Berkeley, CA, pp. 453–484.Google Scholar
Bhattacharya, R. N. (1977). Refinements of the multidimensional central limit theorem and applications. Ann. Prob. 5, 1–27.Google Scholar
Biggins, J. D. (1976). The first- and last-birth problems for a multitype age-dependent branching process. Adv. Appl. Prob. 8, 446–459.Google Scholar
Biggins, J. D. (1992). Uniform convergence of martingales in the branching random walk. Ann. Prob. 20, 137–151.Google Scholar
Biggins, J. D. (1996). How fast does a general branching random walk spread? In Classical and Modern Branching Processes (IMA Vols Math. Appl. 84), eds Athreya, K. B. and Jagers, P.. Springer, New York, pp. 19–40.Google Scholar
Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. Appl. Prob. 36, 544–581.Google Scholar
Biggins, J. D. and Rahimzadeh Sani, A. (2004). Extended Perron–Frobenius results. Preprint 539/04, Department of Probability and Statistics, University of Sheffield. Available at: http://www.shef.ac.uk/∼st1jdb/epfr.html.Google Scholar
Bramson, M., Ney, P. and Tao, J. (1992). The population composition of a multitype branching random walk. Ann. Appl. Prob. 2, 575–596.Google Scholar
Chaganty, N. R. and Sethuraman, J. (1993). Strong large deviation and local limit theorems. Ann. Prob. 21, 1671–1690.Google Scholar
Hörmander, L. (1973). An Introduction to Complex Analysis in Several Variables. North-Holland, Amsterdam.Google Scholar
Jensen, J. L. (1995). Saddlepoint Approximations. Oxford University Press.Google Scholar
Kingman, J. F. C. (1961). A convexity property of positive matrices. Quart. J. Math. Oxford 12, 283–284.Google Scholar
Kontoyiannis, Y. and Meyn, S. P. (2003). Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Prob. 13, 304–362.Google Scholar
Kyprianou, A. E. and Rahimzadeh Sani, A. (2001). Martingale convergence and the functional equation in the multi-type branching random walk. Bernoulli 7, 593–604.Google Scholar
Lancaster, P. and Tismenetsky, M. (1985). The Theory of Matrices, 2nd edn. Academic Press, New York.Google Scholar
Miller, H. D. (1961). A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32, 1260–1270.Google Scholar
Mode, C. J. (1971). Multitype Branching Processes. Theory and Applications. American Elsevier, New York.Google Scholar
Ney, P. and Numelin, E. (1987). Markov additive processes. I. Eigenvalue properties and limit theorems. Ann. Prob. 15, 561–592.Google Scholar
Seneta, E. (1973). Non-Negative Matrices. Allen and Unwin, London.Google Scholar
Stone, C. (1965). A local limit theorem for nonlattice multi-dimensional distribution functions. Ann. Math. Statist. 36, 546–551.Google Scholar
Stone, C. (1967). On local and ratio limit theorems. In Proc. Fifth Berkeley Symp. Math. Statist. Prob. (Berkeley, CA, 1965/66), Vol. 2, University of California Press, Berkeley, CA, pp. 217–224.Google Scholar
Von Bahr, B. (1967). Multi-dimensional integral limit theorems. Ark. Mat. 7, 71–88.Google Scholar
Von Bahr, B. and Esseen, C.-G. (1965). Inequalities for the rth absolute moment of a sum of random variables, 1≤ r≤ 2. Ann. Math. Statist. 36, 299–303.Google Scholar