Large deviations in the supercritical branching process. (English) Zbl 0796.60090

Biggins, J. D.; Bingham, N. H.

Adv. Appl. Probab. 25, No. 4, 757-772 (1993).

Let \(W\) be the limit of the normed supercritical simple branching process. The functional equation satisfied by the Laplace transform of \(W\) has been used to obtain properties of the distribution of \(W\). Detailed study of the left and right hand tails was initiated in a pioneering paper of T. E. Harris [Ann. Math. Stat. 19, 474-494 (1948; Zbl 0041.456)], continued in work of M. S. Dubuc [Z. Wahrscheinlichkeitstheorie Verw. Geb. 19, 281-290 (1971; Zbl 0208.442)], and more recently stimulated by N. H. Bingham [J. Appl. Probab., Spec. Vol. 25A, 215-228 (1988; Zbl 0669.60078)]. The present paper makes further use of existing analytic results to obtain more detailed information about the left and right hand tails, under various conditions on the family size distribution.

Reviewer: D.R.Grey (Sheffield)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60F10	Large deviations

Keywords:

tail behaviour; supercritical simple branching process; family size distribution

Citations:

Zbl 0215.256; Zbl 0041.456; Zbl 0208.442; Zbl 0669.60078