Geodesic
Conditional $L^1$-Convergence for the Martingale of a Critical Branching Process in Random Environment
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 195-206.

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For a critical branching process $(Z_n)$ in a random environment $(\xi _n)$, a sufficient condition is given for the corresponding martingale ${Z_n}/{e^{S_n}}$ to converge in $L^1$ or to degenerate under $\mathbb P^+$, the probability under which the associated random walk is conditioned to stay nonnegative.
Keywords: branching process, random environment, multitype branching processes, change of measure
Mots-clés : martingale convergence. 
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     author = {Wenming Hong and Shengli Liang and Xiaoyue Zhang},
     title = {Conditional $L^1${-Convergence} for the {Martingale} of a {Critical} {Branching} {Process} in {Random} {Environment}},
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Wenming Hong; Shengli Liang; Xiaoyue Zhang. Conditional $L^1$-Convergence for the Martingale of a Critical Branching Process in Random Environment. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 195-206. http://geodesic.mathdoc.fr/item/TM_2022_316_a12/

[1] Afanasyev V.I., Geiger J., Kersting G., Vatutin V.A., “Criticality for branching processes in random environment”, Ann. Probab., 33:2 (2005), 645–673 | DOI | MR | Zbl

[2] Athreya K.B., “Change of measures for Markov chains and the $L\log L$ theorem for branching processes”, Bernoulli, 6:2 (2000), 323–338 | DOI | MR | Zbl

[3] Athreya K.B., Karlin S., “On branching processes with random environments. I: Extinction probabilities”, Ann. Math. Stat., 42:5 (1971), 1499–1520 | DOI | MR | Zbl

[4] Athreya K.B., Karlin S., “Branching processes with random environments. II: Limit theorems”, Ann. Math. Stat., 42:6 (1971), 1843–1858 | DOI | MR | Zbl

[5] Biggins J.D., Kyprianou A.E., “Measure change in multitype branching”, Adv. Appl. Probab., 36:2 (2004), 544–581 | DOI | MR | Zbl

[6] Durrett R., Probability: Theory and examples, 5th ed., Cambridge Univ. Press, Cambridge, 2019 | MR | Zbl

[7] W. Feller, An Introduction to Probability Theory and Its Applications, v. 2, J. Wiley Sons, New York, 1971 | MR | MR | Zbl

[8] Geiger J., Kersting G., “The survival probability of a critical branching process in a random environment”, Theory Probab. Appl., 45:3 (2001), 517–525 | DOI | MR

[9] Kallenberg O., Foundations of modern probability, Springer, New York, 2002 | MR | Zbl

[10] Kersting G., Vatutin V., Discrete time branching processes in random environment, J. Wiley Sons, Hoboken, NJ, 2017 | Zbl

[11] Kesten H., Stigum B.P., “A limit theorem for multidimensional Galton–Watson processes”, Ann. Math. Stat., 37:5 (1966), 1211–1223 | DOI | MR | Zbl

[12] M. V. Kozlov, “On the asymptotic behavior of the probability of non-extinction for critical branching processes in a random environment”, Theory Probab. Appl., 21:4 (1977), 791–804 | DOI | MR | Zbl

[13] Lyons R., Pemantle R., Peres Y., “Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes”, Ann. Probab., 23:3 (1995), 1125–1138 | DOI | MR | Zbl

[14] Smith W.L., Wilkinson W.E., “On branching processes in random environments”, Ann. Math. Stat., 40 (1969), 814–827 | DOI | MR | Zbl

[15] Tanaka H., “Time reversal of random walks in one dimension”, Tokyo J. Math., 12:1 (1989), 159–174 | DOI | MR | Zbl

[16] Tanny D., “A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means”, Stoch. Process. Appl., 28:1 (1988), 123–139 | DOI | MR | Zbl

[17] Vatutin V.A., Wachtel V., “Local probabilities for random walks conditioned to stay positive”, Probab. Theory Relat. Fields, 143:1–2 (2009), 177–217 | DOI | MR | Zbl