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On tails of fixed points of the smoothing transform in the boundary case. (English) Zbl 1183.60031
Let \(\{A_i\}\) be a sequence of positive random numbers such that only the first \(N\) of them are strictly positive, \(N\) being an a.s. finite random integer. Consider the nonnegative solutions of the distribution equation \(Z=_d\sum^N_{i=1} A_i Z_i\), where \(Z,Z_1,Z_2,\dots\) are i.i.d. and independent of \(N,A_1,A_2,\dots\) . If \({\mathbf E} \sum^N_{i= 1} A_i= 1\) and \({\mathbf E}\sum^N_{i=1} A_i\log A_i= 0\), it is known that solutions exist and that they all have infinite mean. Assuming in addition that \({\mathbf E}\sum^N_{i=1} A^{1-\delta}_i< \infty\) for some \(\delta> 0\), \({\mathbf E}(\sum^N_{i=1} JA_i)^{1+ \varepsilon}< \infty\) for some \(\varepsilon> 0\), and \(1<{\mathbf E} N\leq\infty\), the author proves the following:
Let \(Z\) be a nonnegative, non-zero solution. If the \(A_i\) are aperiodic (i.e., there is no positive number \(h\) such that \(\log A_i\) is a.s. an integer multiple of \(h\), for \(1\leq i\leq N\)), then \(\lim_{x\to\infty} x{\mathbf P}(Z> x)= C_0\) for some finite strictly positive number \(C_0\). If the \(A_i\) are periodic, then there exist two positive constants \(C_1\) and \(C_2\) such that \(C_1=\liminf_{x\to\infty} x{\mathbf P}(Z> x)\leq \limsup_{x\to\infty} x{\mathbf P}(Z> x)= C_2\). The problem is reduced to the study of the behaviour at infinity of an invariant measure of a random difference equation, see Y. Guivarc’h [Ann. Inst. Henri Poincaré, Probab. Stat. 26, No. 2, 261–285 (1990; Zbl 0703.60012)] and Q. Liu [Stochastic Processes Appl. 86, No. 2, 263–286 (2000; Zbl 1028.60087)].

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G42 Martingales with discrete parameter
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[1] Babillot, M.; Bougerol, P.; Élie, L., The random difference equation \(X_n = A_n X_{n - 1} + B_n\) in the critical case, Ann. probab., 25, 1, 478-493, (1997) · Zbl 0873.60045
[2] Biggins, J.D.; Kyprianou, A.E., Seneta – heyde norming in the branching random walk, Ann. probab., 25, 337-360, (1997) · Zbl 0873.60062
[3] Biggins, J.D.; Kyprianou, A.E., Fixed points of the smoothing transform: the boundary case, Electron. J. probab., 10, 17, 609-631, (2005) · Zbl 1110.60081
[4] S. Brofferio, D. Buraczewski, E. Damek, On the invariant measure of the random difference equation \(X_n = A_n X_{n - 1} + B_n\) in the critical case, preprint, arxiv.org/abs/0809.1864 · Zbl 1259.60077
[5] Buraczewski, D., On invariant measures of stochastic recursions in a critical case, Ann. appl. probab., 17, 4, 1245-1272, (2007) · Zbl 1151.60034
[6] Durret, R.; Liggett, T.M., Fixed points of the smoothing transformation, Z. wahrscheinlichkeitstheor. verwandte geb., 64, 3, 275-301, (1983) · Zbl 0506.60097
[7] Goldie, C.M., Implicit renewal theory and tails of solutions of random equations, Ann. appl. probab., 1, 1, 126-166, (1991) · Zbl 0724.60076
[8] Guivarc’h, Y., Sur une extension de la notion de loi semi-stable, Ann. inst. H. Poincaré probab. statist., 26, (1990) · Zbl 0703.60012
[9] Kesten, H., Random difference equations and renewal theory for products of random matrices, Acta math., 131, 207-248, (1973) · Zbl 0291.60029
[10] Liu, Q., Fixed points of a generalized smoothing transform and applications to the branching processes, Adv. in appl. probab., 30, 85-112, (1998) · Zbl 0909.60075
[11] Liu, Q., Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks, Stochastic process. appl., 82, 61-87, (1999) · Zbl 0997.60091
[12] Liu, Q., On generalized multiplicative cascades, Stochastic process. appl., 86, 2, 263-286, (2000) · Zbl 1028.60087
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