, Volume 71, Issue 1, pp 85-116

A weighted occupation time for a class of measured-valued branching processes

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A weighted occupation time is defined for measure-valued processes and a representation for it is obtained for a class of measure-valued branching random motions on R d. Considered as a process in its own right, the first and second order asymptotics are found as time t→∞. Specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d, and when infinite, a central limit type renormalization is considered, yielding Gaussian or asymmetric stable generalized random fields in the limit. In one Gaussian case the results are contrasted in high versus low dimensions.