Take any subcritical offspring distribution with bounded support and consider the corresponding Galton--Watson tree. In this short note we condition this Galton--Watson tree on large width and show that the conditioned tree does not converge locally to any random tree with at most one infinite spine.
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HE Xin. A Counterexample on Local Limits of Galton--Watson Trees. CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST, 2018, 34(1): 1-7.