Abstract
Let ( Z n ) n⩾0 be a branching process in random environment represented by a sequence of i.i.d. generating functions ( f n ) n⩾0 . In the subcritical case, Elog f′(1)<0, the non-extinction probability at generation n decays exponentially fast, the rate depending on whether E[ f′(1)log f′(1)] is less, equal or greater than 0. We determine the exact asymptotic of the non-extinction probability P( Z n >0) in all three cases under suitable integrability assumptions. Moreover, we show that Z n conditioned on Z n >0 has a non-degenerate limit law.
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