Convergence in \(L^p\) and its exponential rate for a branching process in a random environment

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Abstract: We consider a supercritical branching process

(Z_{n})

in a random environment

x i

. Let

W

be the limit of the normalized population size

W_{n} = Z_{n} / E [Z_{n} | x i]

. We first show a necessary and sufficient condition for the quenched

L^{p}

(

p > 1

) convergence of

(W_{n})

, which completes the known result for the annealed

L^{p}

convergence. We then show that the convergence rate is exponential, and we find the maximal value of

h o > 1

such that

h o^{n} (W - W_{n}) i g h t a r r o w 0

in

L^{p}

, in both quenched and annealed sense. Similar results are also shown for a branching process in a varying environment.

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