  Consider~\(n\) nodes~\(\{X_i\}_{1 \leq i \leq n}\) independently distributed
in the unit square~\(S,\) each according to a distribution~\(f.\) Nodes~\(X_i\)
and~\(X_j\) are joined by an edge if the Euclidean distance~\(d(X_i,X_j)\) is
less than~\(r_n,\) the adjacency distance and the resulting random
graph~\(G_n\) is called a random geometric graph~(RGG). We now assign a
location dependent weight to each edge of~\(G_n\) and define~\(MST_n\) to be
the sum of the weights of the minimum spanning trees of all components
of~\(G_n.\) For values of~\(r_n\) above the connectivity regime, we obtain
upper and lower bound deviation estimates for~\(MST_n\) and~\(L^2-\)convergence
of~\(MST_n\) appropriately scaled and centred.
