A necessary and sufficient condition for noncertain extinction of a branching process in a random environment (BPRE) |
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Authors: | John Coffey David Tanny |
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Institution: | Department of Mathematics, State University College at Geneseo, Geneseo, NY 14454, U.S.A.;Department of Mathematics, York University, Downsview, Ont., Canada M3J 1P3, Canada |
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Abstract: | It is known that a branching process in a random environment (BPRE) which is subcritical or critical either dies with probability one or, in the trivial case, corresponds to an immortal sterile population. In the supercritical case, various conditions are known to be necessary for noncertain extinction while other conditions are known to be sufficient. In this paper, a necessary and sufficient condition for noncertain extinction of a supercritical BPRE is given. In particular, it is shown that a supercritical BPRE has noncertain extinction if and only if there exists a random truncation, depending only on the environmental sequence, such that the truncated BPRE is supercritical and such that the sequence of truncation points grows more slowly than any exponential sequence. |
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Keywords: | Primary 60J80 Secondary 60F15 |
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