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Criticality for branching processes in random environment. (English) Zbl 1075.60107
Let $Z$ be a critical, discrete-time, one-type Markov branching process in a random environment $\Pi= (Q_1, Q_2, Q_3,\dots)$, $Q_k$ i.i.d copies of $Q$, $Q$ having probability measures on $\{0,1,2,\dots\}$ as its values. Define $m(Q):= \sum_{y> 0}yQ(\{y\})$, $\zeta(a):= \sum_{y\ge a}y^2Q(\{y\})/m(Q)^2$, $a= 0,1,2,\dots$, let $S =(S_0, S_1,S_2,\dots)$ be the random walk given by $X_n= S_n- S_{n-1}:= \log m(Q_n)$. Suppose $X_n$ is a.s. finite and define $v(x):= 1+ \sum_{i> 1}{\bold P}(S_{\gamma(i)}\ge -x)$ when $x\ge 0$, and $v(x):= 0$ otherwise, where the $\gamma(i)$ are the strict descending ladder epochs of $S$. Set $X^{r,n}_t:= Z_{r+[(n- r)t]}/\mu_{r+[(n- r)t]}$, $0\le t< 1$, where $\mu_n:= {\bold E}(Z_n, Z_0,\Pi)$. Consider two sets of assumptions: (A) There exits a $\rho$, $0< \rho< 1$, such that $(1/n)\sum_{1\le m\le n}{\bold P}(S_m> 0)<\rho$, $n\to \infty$ and, for some $\varepsilon> 0$ and some integer $a> 0$, ${\bold E}(\log^+(\zeta(a))^{1/\rho+\varepsilon}<\infty$ and ${\bold E}[v(X_1)(\log^+\zeta(a)^{1+\varepsilon}]<\infty$. (B) The distribution of $X_1$ belongs without centering to the domain of attraction of some stable law which is not one-sided and has index $\alpha$, $0<\alpha\le 2$, and, for some $\varepsilon> 0$ and some integer $a> 0$, ${\bold E}(\log^+\zeta(a))^{\alpha+\varepsilon}< \infty$. First, assume (A) or (B). Then, for some finite positive real number $\theta$, $${\bold P}(Z_n> 0)\sim\theta{\bold P}(\min(S_1,S_2,\dots,S_n)\ge 0), \quad n\to\infty,$$ and, as a corollary, $${\bold P}(Z_n> 0)\sim\theta n^{-(1-\rho)}l(n), \quad n\to\infty,$$ with $l(n)$ slowly varying at infinity. For any sequence of integers $r(1),r(2),r(3),\dots$ such that $r(n)< n$ and $r(n)\to\infty$, as $n\to \infty$, the conditional distribution of $(X^{r(n),n}\mid Z_n> 0)$ converges weakly (Skorokhod topology) to the law of a process with a.s. constant paths, $0< W_t<\infty$ a.s.. The distribution of $(\min\{i\le n: S_i= \min(S_0,\dots, S_n)\},\min(S_0,\dots, S_n)\mid Z_n> 0)$ converges weakly to a probability measure. Assuming (B), there exists a sequence $l(1),l(2),\dots$ slowly varying at infinity, such that the conditional distribution of $((n^{-1/\alpha}l(n) S_{[nt]})_{0\le t\le 1}\mid Z_n> 0)$ converges weakly to the law of the meander of a strictly stable process with index, and, as a corollary, the corresponding result for $\log Z_{[nt]}$ in place of $S_{[nt]}$ holds.

MSC:
60J80Branching processes
60G50Sums of independent random variables; random walks
60F17Functional limit theorems; invariance principles
60K37Processes in random environments
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References:
[1] Afanasyev, V. I. (1993). A limit theorem for a critical branching process in random environment. Discrete Math. Appl. 5 45--58. (In Russian.) · Zbl 0803.60082
[2] Afanasyev, V. I. (1997). A new theorem for a critical branching process in random environment. Discrete Math. Appl. 7 497--513. · Zbl 0969.60085 · doi:10.1515/dma.1997.7.5.497
[3] Afanasyev, V. I. (2001). A functional limit theorem for a critical branching process in a random environment. Discrete Math. Appl. 11 587--606. · Zbl 1102.60306 · doi:10.1515/dma.2001.11.6.587
[4] Agresti, A. (1975). On the extinction times of varying and random environment branching processes. J. Appl. Probab. 12 39--46. · Zbl 0306.60052 · doi:10.2307/3212405
[5] Athreya, K. B. and Karlin, S. (1971). On branching processes with random environments: I, II. Ann. Math. Statist. 42 1499--1520, 1843--1858. · Zbl 0228.60032 · doi:10.1214/aoms/1177693150
[6] Athreya, K. B. and Ney, P. (1972). Branching Processes . Springer, Berlin. · Zbl 0259.60002
[7] Bertoin, J. and Doney, R. A. (1994). On conditioning a random walk to stay nonnegative. Ann. Probab. 22 2152--2167. JSTOR: · Zbl 0834.60079 · doi:10.1214/aop/1176988497 · http://links.jstor.org/sici?sici=0091-1798%28199410%2922%3A4%3C2152%3AOCARWT%3E2.0.CO%3B2-S&origin=euclid
[8] Bingham, N., Goldie, C. and Teugels, J. (1987). Regular Variation . Cambridge Univ. Press. · Zbl 0617.26001
[9] Doney, R. A. (1985). Conditional limit theorems for asymptotically stable random walks. Z. Wahrsch. Verw. Gebiete 70 351--360. · Zbl 0573.60063 · doi:10.1007/BF00534868
[10] Doney, R. A. (1995). Spitzer’s condition and ladder variables in random walks. Probab. Theory Related Fields 101 577--580. · Zbl 0818.60060 · doi:10.1007/BF01202785
[11] Durrett, R. (1978). Conditioned limit theorems for some null recurrent Markov processes. Ann. Probab. 6 798--828. JSTOR: · Zbl 0398.60023 · doi:10.1214/aop/1176995430 · http://links.jstor.org/sici?sici=0091-1798%28197810%296%3A5%3C798%3ACLTFSN%3E2.0.CO%3B2-C&origin=euclid
[12] Dyakonova, E. E., Geiger, J. and Vatutin, V. A. (2004). On the survival probability and a functional limit theorem for branching processes in random environment. Markov Process. Related Fields 10 289--306. · Zbl 1078.60065
[13] Feller, W. (1971). An Introduction to Probability Theory and Its Applications II . Wiley, New York. · Zbl 0219.60003
[14] Geiger, J. and Kersting, G. (2000). The survival probability of a critical branching process in a random environment. Theory Probab. Appl. 45 517--525. · Zbl 0994.60095 · doi:10.1137/S0040585X97978440
[15] Jagers, P. (1974). Galton--Watson processes in varying environments. J. Appl. Probab. 11 174--178. · Zbl 0277.60061 · doi:10.2307/3212594
[16] Jirina, M. (1976). Extinction of non-homogeneous Galton--Watson processes. J. Appl. Probab. 13 132--137. · Zbl 0365.60087 · doi:10.2307/3212673
[17] Kozlov, M. V. (1976). On the asymptotic behavior of the probability of non-extinction for critical branching processes in a random environment. Theory Probab. Appl. 21 791--804. · Zbl 0384.60058 · doi:10.1137/1121091
[18] Kozlov, M. V. (1995). A conditional function limit theorem for a critical branching process in a random medium. Dokl. Akad. Nauk 344 12--15. (In Russian.) · Zbl 0884.60085
[19] Petrov, V. V. (1975). Sums of Independent Random Variables . Springer, Berlin. · Zbl 0322.60043
[20] Smith, W. L. and Wilkinson, W. E. (1969). On branching processes in random environments. Ann. Math. Statist. 40 814--827. · Zbl 0184.21103 · doi:10.1214/aoms/1177697589
[21] Tanaka, H. (1989). Time reversal of random walks in one dimension. Tokyo J. Math. 12 159--174. · Zbl 0692.60052 · doi:10.3836/tjm/1270133555
[22] Vatutin, V. A. (2002). Reduced branching processes in random environment: The critical case. Theory Probab. Appl. 47 99--113. · Zbl 1039.60077 · doi:10.1137/S0040585X97979421
[23] Vatutin, V. A. and Dyakonova, E. E. (2003). Galton--Watson branching processes in random environment, I: Limit theorems. Teor. Veroyatnost. i Primenen. 48 274--300. (In Russian.) · Zbl 1079.60080 · doi:10.1137/S0040585X97980373