×

Revisiting John Lamperti’s maximal branching process. (English) Zbl 1492.60238

Summary: Lamperti’s maximal branching process is revisited, with emphasis on the description of the shape of the invariant measures in both the recurrent and transient regimes. A truncated version of this chain is exhibited, preserving the monotonicity of the original Lamperti chain supported by the integers. The Brown theory of hitting times applies to the latter chain with finite state-space, including sharp strong time to stationarity. Additional information on these hitting time problems are drawn from the quasi-stationary point of view.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aydogmus, O.; Ghosh, A. P.; Ghosh, S.; Roitershtein, A., Coloured maximal branching process, Theory Probab. Appl., 59, 4, 663-672 (2015) · Zbl 1376.60073
[2] Brown, M., Consequences of monotonicity for Markov transition functions, City College, CUNY Report No MB89-03, unpublished manuscript, 1990.
[3] Brown, M., Error bounds for exponential approximations of geometric convolutions, Ann. Probab., 18, 3, 1388-1402 (1990) · Zbl 0709.60016
[4] Collet, P., Martínez, S., and San Martín, J., Quasi-Stationary Distributions, Probability and Its Applications, Springer, New York, 2013. · Zbl 1261.60002
[5] Comtet, L., Analyse Combinatoire, Tomes 1 et 2. Presses Universitaires de France, Paris, 1970. · Zbl 0221.05002
[6] Embrechts, P., Klüppelberg, C., and Mikosch, T., Modelling Extremal Events for Insurance and Finance, Stochastic Modelling and Applied Probability, Springer, Berlin, Heidelberg, New-York, 1997. · Zbl 0873.62116
[7] Flajolet, P.; Odlyzko, A., Singularity analysis of generating functions, SIAM J. Discrete Math., 3, 2, 216-240 (1990) · Zbl 0712.05004
[8] Foster, F. G., On the stochastic matrices associated with certain queuing processes, Ann. Math. Statist., 24, 355-360 (1953) · Zbl 0051.10601
[9] Grassmann, W. K.; Heyman, D. P., Equilibrium distribution of block-structured Markov chains with repeating rows, J. Appl. Probab., 27, 3, 557-576 (1990) · Zbl 0716.60076
[10] Gupta, P. L.; Gupta, R. C.; Tripathi, R. C., On the monotonic properties of discrete failure rates, J. Statist. Plann. Inference, 65, 255-268 (1997) · Zbl 0908.62099
[11] Harris, T.E., The Existence of Stationary Measures for Certain Markov Processes, Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, Vol. II. University of California Press, Berkeley and Los Angeles, 1956, pp. 113-124. · Zbl 0072.35201
[12] Harris, T. E., Transient Markov chains with stationary measures, Proc. Amer. Math. Soc., 8, 937-942 (1957) · Zbl 0087.13501
[13] Keilson, J.; Kester, A., Monotone matrices and monotone Markov processes, Stoch. Processes Their Appl., 5, 231-241 (1977) · Zbl 0367.60078
[14] Lamperti, J., Maximal branching processes and ‘long-range percolation’, J. Appl. Probab., 7, 1, 89-98 (1970) · Zbl 0196.18801
[15] Lamperti, J., Remarks on maximal branching processes, Teor. Verojatnost. I Primenen., 17, 46-54 (1972) · Zbl 0279.60078
[16] Lebedev, A. V., A double exponential law for maximal branching processes, Discrete Math. Appl., 12, 4, 415-420 (2002) · Zbl 1045.60091
[17] Lebedev, A. V., Generalized maximal branching processes in the case of power tails, Moscow Univ. Math. Bull., 60, 2, 34-36 (2005) · Zbl 1103.60065
[18] Lebedev, A. V., Maximal branching processes with nonnegative values, Theory Probab. Appl., 50, 3, 482-488 (2006) · Zbl 1115.60079
[19] Lebedev, A. V., Tail behavior of the stationary distributions of the maximal branching processes, Theory Probab. Appl., 54, 4, 699-702 (2010) · Zbl 1222.60063
[20] Lebedev, A. V., Maximal branching processes with several types of particles, Moscow Univ. Math. Bull., 67, 3, 97-101 (2012) · Zbl 1272.60062
[21] Lorek, P., Speed of convergence to stationarity for stochastically monotone Markov chains, Ph.D. diss., Mathematical Institute, University of Wrowclaw, 2007.
[22] Pitman, J. and Tang, W., Tree formulas, mean first passage times and Kemenys constant of a Markov chain, preprint (2016). arXiv:1603.09017.
[23] Pollak, M.; Siegmund, D., Convergence of quasistationary to stationary distributions for stochastically monotone Markov processes, J. Appl. Probab., 23, 1, 215-220 (1986) · Zbl 0595.60075
[24] Sibuya, M., Generalized hypergeometric, digamma and trigamma distributions, Ann. Inst. Statist. Math., 31, 373-390 (1979) · Zbl 0448.62008
[25] Steutel, F.W. and van Harn, K., Infinite divisibility of probability distributions on the real line, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 259, Marcel Dekker, Inc., New York, 2004. · Zbl 1063.60001
[26] Zhao, Y. Q.; Liu, D., The censored Markov chain and the best augmentation, J. Appl. Probab., 33, 3, 623-629 (1996) · Zbl 0865.60007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.