  We consider linear preferential attachment random trees with additive
fitness, where fitness is defined as the random initial vertex attractiveness.
We show that when the fitness distribution has positive bounded support, the
weak local limit of this family can be constructed using a sequence of mixed
Poisson point processes. We also provide a rate of convergence of the total
variation distance between the r- neighbourhood of the uniformly chosen vertex
in the preferential attachment tree and that of the root vertex of its weak
local limit. We apply the theorem to obtain the limiting degree distributions
of the uniformly chosen vertex and its ancestors, that is, the vertices that
are on the path between the uniformly chosen vertex and the initial vertex.
Rates of convergence in the total variation distance are established for these
results.

Probability