Minimum spanning trees of random geometric graphs with location dependent weights

Consider n nodes {Xi}1≤i≤n independently distributed in the unit square S, each according to a distribution f. Nodes Xi and Xj are joined by an edge if the Euclidean distance d(Xi,Xj) is less than rn, the adjacency distance and the resulting random graph Gn is called a random geometric graph (RGG). We now assign a location dependent weight to each edge of Gn and define MSTn to be the sum of the weights of the minimum spanning trees of all components of Gn. For values of rn above the connectivity regime, we obtain upper and lower bound deviation estimates for MSTn and L2-convergence of MSTn appropriately scaled and centred.