A. For, . Number, and . Parasites, This gives the upper bound by Markov inequality and completes the proof Functional limit theorems for strongly subcritical branching processes in random environment. Stochastic Process, Appl, vol.115, issue.2 10, pp.1658-1676, 2005.

V. I. Afanasyev, J. Geiger, G. Kersting, and V. A. Vatutin, Branching processes in random environment. A phase transition in the subcritical regime

V. I. Afanasyev, J. Geiger, G. Kersting, and V. Vatutin, Criticality for branching processes in random environment, The Annals of Probability, vol.33, issue.2, pp.645-673, 2005.
DOI : 10.1214/009117904000000928

A. Agresti, On the extinction times of varying and random environment branching processes, Journal of Applied Probability, vol.12, issue.01, pp.39-46, 1975.
DOI : 10.2307/1426296

S. Asmussen and H. Hering, Branching processes, Progress in Probability and Statistics, 1983.

K. B. Athreya and S. Karlin, On Branching Processes with Random Environments: I: Extinction Probabilities, The Annals of Mathematical Statistics, vol.42, issue.5, pp.1499-1520, 1971.
DOI : 10.1214/aoms/1177693150

K. B. Athreya and S. Karlin, Branching Processes with Random Environments, II: Limit Theorems, The Annals of Mathematical Statistics, vol.42, issue.6, pp.1843-1858, 1971.
DOI : 10.1214/aoms/1177693051

K. B. Athreya and P. E. Ney, Branching processes, 2004.
DOI : 10.1007/978-3-642-65371-1

K. Athreya and H. J. Kang, Some limit theorems for positive recurrent branching Markov chains: I, Advances in Applied Probability, vol.2, issue.03, pp.693-710, 1998.
DOI : 10.1016/0022-247X(67)90155-2

K. Athreya and H. J. Kang, Some limit theorems for positive recurrent branching Markov chains: II, Advances in Applied Probability, vol.30, issue.03, pp.711-722, 1998.
DOI : 10.1090/S0002-9947-1978-0511425-0

URL : http://projecteuclid.org/download/pdf_1/euclid.aap/1035228125

V. Bansaye, On a model for the storage of files on a hardware I : Statistics at a fixed time and asymptotics. Preprint avialable via http, p.611432, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00113823

V. Bansaye, On a Model for the Storage of Files on a Hardware. II. Evolution of a Typical Data Block, Journal of Applied Probability, vol.95, issue.04, pp.901-927, 2007.
DOI : 10.1090/S0002-9947-1977-0433606-6

URL : https://hal.archives-ouvertes.fr/hal-00119860

V. Bansaye, Limit theorems for subcritical branching processes in random environment depending on the initial number of particles. Preprint available via http, p.853, 2008.

V. Bansaye, Cell contamination and branching processes in random environment with immigration. Preprint available via http, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00485290

A. D. Barbour and M. Kafetzaki, A host-parasite model yielding heterogeneous parasite loads, Journal of Mathematical Biology, vol.31, issue.2, pp.157-176, 1993.
DOI : 10.1007/BF00171224

A. D. Barbour and M. J. Luczak, Laws of large numbers for epidemic models with countably many types, The Annals of Applied Probability, vol.18, issue.6, 2006.
DOI : 10.1214/08-AAP521

I. Benjamini and Y. Peres, Markov Chains Indexed by Trees, The Annals of Probability, vol.22, issue.1, pp.219-243, 1994.
DOI : 10.1214/aop/1176988857

J. Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol.121, 1996.

J. Bertoin, Subordinators: examples and applications. Lectures on probability theory and statistics (Saint-Flour), Lecture Notes in Math, 1717.
DOI : 10.1007/978-3-540-48115-7_1

URL : http://www.zora.uzh.ch/79481/1/M2-Bertoin-subordinateurs.pdf

J. Bertoin, Subordinators, Lévy processes with no negative jumps and branching processes. Lecture notes for MaPhySto Available via http, 2000.
DOI : 10.1007/bf01195886

J. Bertoin, Eternal solutions to Smoluchowski's coagulation equation with additive kernel and their probabilistic interpretations, The Annals of Applied Probability, vol.12, issue.2, pp.547-564, 2002.
DOI : 10.1214/aoap/1026915615

J. Bertoin and G. Miermont, Asymptotics in Knuth's parking problem for caravans, Random Structures and Algorithms, vol.3, issue.1, pp.38-55, 2006.
DOI : 10.1002/rsa.20092

P. Billingsley, Convergence of probability measures, 1999.
DOI : 10.1002/9780470316962

N. H. Bingham, On the limit of a supercritical branching process, Journal of Applied Probability, vol.5, issue.A, pp.215-228, 1988.
DOI : 10.2307/1426407

N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation. Encyclopedia of Mathematics and its Applications, 27, 1989.

O. J. Boxma and J. W. Cohen, Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions. Queueing Systems Theory Appl, pp.177-204, 1999.

P. Chassaing and G. Louchard, Phase transition for Parking blocks, Brownian excursion and coalescence, Random Structures and Algorithms, vol.7, issue.1, pp.76-119, 2002.
DOI : 10.1002/rsa.10039

URL : https://hal.archives-ouvertes.fr/hal-00003777

P. Chassaing and P. Flajolet, Hachage, arbres, chemins. Gazette des mathématiciens, 2003.

J. W. Cohen, The single server queue, 1982.

F. M. Dekking, On the survival probability of a branching process in a finite state iid environment. Stochastic Process, Appl, vol.27, pp.151-157, 1988.

S. Dubuc, La densit??? de la loi-limite d'un processus en cascade expansif, Zeitschrift f???r Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol.37, issue.4, pp.281-290, 1971.
DOI : 10.1007/BF00535833

S. Dubuc, Probl??mes relatifs ?? l'it??ration de fonctions sugg??r??s par les processus en cascade, Annales de l???institut Fourier, vol.21, issue.1, pp.171-251, 1971.
DOI : 10.5802/aif.365

URL : http://archive.numdam.org/article/AIF_1971__21_1_171_0.pdf

R. Durrett, Conditioned limit theorems for random walks with negative drift, Zeitschrift f???r Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol.10, issue.3, pp.277-287, 1980.
DOI : 10.1007/BF00538892

S. N. Evans and D. Steinsaltz, Damage segregation at fissioning may increase growth rates : a superprocess model Avialable via http, 2006.

W. Feller, An introduction to probability theory and its applications, 1971.

P. Flajolet, P. Poblete, and A. Viola, On the Analysis of Linear Probing Hashing, Algorithmica, vol.22, issue.4, pp.490-515, 1998.
DOI : 10.1007/PL00009236

URL : https://hal.archives-ouvertes.fr/inria-00073424

K. Fleischmann and V. Vatutin, Reduced subcritical Galton-Watson processes in a random environment, Advances in Applied Probability, vol.27, issue.01, pp.88-111, 1999.
DOI : 10.2307/1427834

K. Fleischmann and V. , On the left tail asymptotics for the limit law of supercritical Galton-Watson processes in the Böttcher case Avialable via http, 2007.

P. J. Fitzsimmons, B. Fristedt, and B. Maisonneuve, Intersections and limits of regenerative sets, Zeitschrift f??r Wahrscheinlichkeitstheorie und verwandte Gebiete, vol.63, issue.2, pp.157-173, 1985.
DOI : 10.1007/BF02451426

D. Foata and J. Riordan, Mappings of acyclic and parking functions, Aequationes Mathematicae, vol.4, issue.1, pp.10-22, 1974.
DOI : 10.1007/BF01834776

J. Geiger, G. Kersting, and V. A. Vatutin, Th??or??mes limites pour des processus de branchement sous-critiques en environnement al??atoire, Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol.39, issue.4, pp.593-620, 2003.
DOI : 10.1016/S0246-0203(02)00020-1

P. Greenwood and J. Pitman, Construction of Local Time and Poisson Point Processes from Nested Arrays, Journal of the London Mathematical Society, vol.2, issue.1, pp.182-192, 1980.
DOI : 10.1112/jlms/s2-22.1.182

D. R. Grey and L. Zhunwei, The asymptotic behaviour of extinction probability in the Smith Wilkinson branching processes, Adv. Appl. Prob, vol.25, issue.2, pp.263-289, 1993.

Y. Guivarc-'h and Q. Liu, Propri??t??s asymptotiques des processus de branchement en environnement al??atoire, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.332, issue.4, pp.339-344, 2001.
DOI : 10.1016/S0764-4442(00)01783-3

J. Guyon, Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging, The Annals of Applied Probability, vol.17, issue.5/6, pp.5-6, 2007.
DOI : 10.1214/105051607000000195

P. Haccou, J. Jagers, and V. A. Vatutin, Branching Processes: Variation, Growth and Extinction of Populations, 2005.
DOI : 10.1017/CBO9780511629136

B. Hambly, On the limiting distribution of a supercritical branching process in a random environment, Journal of Applied Probability, vol.25, issue.03, pp.499-518, 1992.
DOI : 10.1007/BF00535833

K. Hirano, Determination of the limiting coefficient for exponentials functional of random walks with positive drift, J. Math. Sci. Univ. Tokyo, vol.5, pp.299-332, 1998.

B. Hambly, On the limiting distribution of a supercritical branching process in a random environment, Journal of Applied Probability, vol.25, issue.03, pp.499-518, 1992.
DOI : 10.1007/BF00535833

J. Hawkes, Tree generated by a simple branching process, J. London Math. Soc, vol.24, issue.2 2, pp.373-384, 1981.

D. L. Iglehart, Conditioned limit theorems for random walks Stochastic processes and related topics, Proc. Summer Res. Inst. Statist. Inference for Stochastic Processes, pp.167-194, 1974.

J. Jacod and A. N. Shiryaev, Limit theorems for stochastic processes, Fundamental Principles of Mathematical Sciences], vol.288, 2003.
DOI : 10.1007/978-3-662-02514-7

O. Kallenberg, Foundations of modern probability, 2002.
DOI : 10.1007/978-1-4757-4015-8

H. Kaspi, M. Rubinovitch, and C. Monogr, Regenerative sets and their applications to Markov storage systems. Queueing theory and its applications, pp.413-427, 1988.

D. Kendall, Unitary dilations of Markov transition operators, and the corresponding integral representations for transition-probability matrices . Probability and statistics : The Harald Cramér volume, pp.139-161, 1959.

E. S. Key, Limiting Distributions and Regeneration Times for Multitype Branching Processes with Immigration in a Random Environment, The Annals of Probability, vol.15, issue.1, pp.344-353, 1987.
DOI : 10.1214/aop/1176992273

M. Kimmel, Quasistationarity in a branching model of division-withindivision . Classical and modern branching processes, Math. Appl, vol.84, pp.157-164, 1994.

M. Kimmel, Branching processes for biology, 2002.
DOI : 10.1007/978-1-4939-1559-0

J. F. Kingman, The heavy traffic approximation in the theory of queues, Proc. Sympos. Congestion Theory (Chapel Hill, pp.137-169, 1964.

J. F. Kingman, Poisson processes. Oxford Studies in Probability, 3, 1993.

M. V. Kozlov, On the asymptotic behaviour of the probability of nonextinction for critical branching processes in a random environment, Theory Probab. Appl, vol.21, issue.4, pp.742-751, 1976.

M. V. Kozlov, On large deviations of branching processes in a random environment: geometric distribution of descendants, Discrete Mathematics and Applications, vol.16, issue.2, pp.29-47, 2006.
DOI : 10.1515/156939206777344593

A. Lambert, Population Dynamics and Random Genealogies, Memorias del IX Simposio de probabilidad y procesos estocásticos, 2006.
DOI : 10.1080/15326340802437728

URL : https://hal.archives-ouvertes.fr/hal-00355166

Q. Liu, On generalized multiplicative cascades. Stochastic Process, Appl, vol.86, issue.2, pp.263-286, 2000.
DOI : 10.1016/s0304-4149(99)00097-6

URL : http://doi.org/10.1016/s0304-4149(99)00097-6

C. J. Luchsinger, Approximating the long-term behaviour of a model for parasitic infection, Journal of Mathematical Biology, vol.42, issue.6, pp.555-581, 2001.
DOI : 10.1007/s002850100083

C. J. Luchsinger, Stochastic models of a parasitic infection, exhibiting three basic reproduction ratios, Journal of Mathematical Biology, vol.42, issue.6, pp.532-554, 2001.
DOI : 10.1007/s002850100082

]. R. Lyons, R. Pemantle, and Y. Peres, Conceptual Proofs of $L$ Log $L$ Criteria for Mean Behavior of Branching Processes, The Annals of Probability, vol.23, issue.3, pp.1125-1138, 1995.
DOI : 10.1214/aop/1176988176

R. Lyons and Y. Peres, Probability on Trees and Networks. Available via http, 2005.
DOI : 10.1017/9781316672815

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.398.1383

B. Maisonneuve, Ensembles r???g???n???ratifs de la droite, Zeitschrift f???r Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol.63, issue.4, pp.501-510, 1983.
DOI : 10.1007/BF00533723

G. Materon, Random sets and integral geometry, 1978.

P. A. Meyer, Un Theoreme Sur La Repartition Des Temps Locaux, Lectures Notes in Mathematics, vol.191, pp.209-210, 1971.
DOI : 10.1007/BFb0058861

P. W. Millar, Zero-one laws and the minimum of a Markov process, Transactions of the American Mathematical Society, vol.226, pp.365-391, 1977.
DOI : 10.1090/S0002-9947-1977-0433606-6

P. Mörters and N. R. Shieh, Thin and thick points for branching measure on a Galton???Watson tree, Statistics & Probability Letters, vol.58, issue.1, pp.13-22, 2002.
DOI : 10.1016/S0167-7152(02)00093-7

P. Mörters and M. Ortgiese, Small value probabilities via the branching tree heuristic Bernoulli, pp.277-299, 2008.

P. E. Ney-;-a and . Vidyashankar, Local limit theory and large deviations for supercritical branching processes, Ann. Appl. Probab, vol.14, issue.3, pp.1135-1166, 2004.

K. R. Parthasarathy, Probability measures on metric spaces. Probability and Mathematical Statistics, 1967.

J. Pitman, Stationary excursions, Séminaire de Probabilités XXI, Lecture notes in Math. 1247, pp.289-302, 1986.
DOI : 10.1214/aop/1176995155

J. Pitman, Combinatorial stochastic processes Lecture notes for Saint Flour course, 2002.

N. U. Prabhu, Stochastic storage processes. Queues, insurance risk, dams, and data communication, 1998.

P. Robert, Stochastic networks and queues. Translated from the, 52. Stochastic Modelling and Applied Probability, 2000.
DOI : 10.1007/978-3-662-13052-0

A. Roitershtein, A note on multitype branching processes with immigration in a random environment, The Annals of Probability, vol.35, issue.4, pp.1573-1592, 2007.
DOI : 10.1214/009117906000001015

S. M. Ross, Stochastic Processes, 1983.

A. Rouault, Large deviations and branching processes Pliska Stud, Proceedings of the 9th International Summer School on Probability Theory and Mathematical Statistics, pp.15-38, 1997.

K. Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol.68, 1999.

W. L. Smith and . Wilkinson, On Branching Processes in Random Environments, The Annals of Mathematical Statistics, vol.40, issue.3, pp.814-827, 1969.
DOI : 10.1214/aoms/1177697589

E. J. Stewart, R. Madden, G. Paul, and F. Taddei, Aging and Death in an Organism That Reproduces by Morphologically Symmetric Division, PLoS Biology, vol.297, issue.2, p.45, 2005.
DOI : 10.1371/journal.pbio.0030045.sv001

URL : https://hal.archives-ouvertes.fr/inserm-00080154

M. I. Taksar, Regenerative sets on real line, Lecture Notes in Math, vol.1, pp.437-474, 1980.
DOI : 10.1007/BFb0089508

M. I. Taksar, Stationary Markov sets, Lecture Notes in Math, vol.63, pp.303-340, 1247.
DOI : 10.1007/BF00533723

V. Vatutin, Limit Theorem for an Intermediate Subcritical Branching Process in a Random Environment, Theory of Probability & Its Applications, vol.48, issue.3, pp.481-492, 2004.
DOI : 10.1137/S0040585X97980518