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- We claim that |IDi(q∗ )| is first-order stochastically dominated by the number of nodes in the Poisson branching process with parameter ι(q, q)δ. To prove this, it is sufficient to show that a random variable with distribution Poisson(ι(q, q)δn) first-order stochastically dominates a random variable with distribution Binom(n, ι(q, q)δ), as IDi(q∗ ) is the set of nodes in a branching process with the distribution of offspring first-order stochastically dominated by Binom(n, ι(q, q)δ). By Theorem 1(f) of Klenke and Mattner (2010), this holds if (1 − δq∗ )n ≤ e−ι(q,q)δn . Letting C0 = ι(q, q)δn, we observe that (1 − C0 n )n is increasing in n and converges to e−C0 , so the inequality holds.
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- We want to show that y0 < ∞ with probability one and the probability that y0 = y decays exponentially in y. The (at most K × K) matrix δι(qi, qj)n(qj) has spectral radius at most λ because (ι(qi, qj)δ)ij does. Therefore, by Theorem 2 of Section V.3 of Athreya and Ney (1972), the probability that y0 = ∞ is zero.
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