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JasTonAChair
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Meaningful interpretation for fixed point of a probability generating function
I'm a bit late to the party, but this resource is very helpful - stats.libretexts.org/Bookshelves/Probability_Theory/…. The key insight is that the probability of extinction $q$ for each generation is greater than the last, but less than 1, so $q_n \to q $ as $n \to \infty $.