For d≥3, the joint occupancy counts of a critical branching random walk approach a deterministic multiple of an exponential with Wasserstein rate n^{−(d−2)/(2(d+1))}; for d≥7, centered fluctuations converge to a multivariate Laplace law with rate n^{−(2d−9)/(6(2d+1))}.
Meckes, 2009
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Exponential and Laplace approximation for occupation statistics of branching random walk
For d≥3, the joint occupancy counts of a critical branching random walk approach a deterministic multiple of an exponential with Wasserstein rate n^{−(d−2)/(2(d+1))}; for d≥7, centered fluctuations converge to a multivariate Laplace law with rate n^{−(2d−9)/(6(2d+1))}.