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The conditions are that the volume function, when composed with $\\log$, is regularly varying and that the limit of the uniform probability measure on a large ball exists in the horocompactification. As an application we prove convergence towards a unique IPVT for higher rank symmetric spaces, which solves an open problem of \\cite{MiMe23}. Versions of this theorem are provided for graphs and edge-measured graphs, where a natural parameter $\\xi$ appears. 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