Given a countably infinite group G acting on some space X, an increasing family of finite subsets Gn, x ∈ X and a function f over X we con- sider the sums Sn(f, x) = g∈Gn f (gx). The asymptotic behaviour of Sn(f, x) is a delicate problem that was studied under various settings. In the following paper we study this problem when G is a specific lattice in SL (2,) acting on the projective line and Gn are chosen using the word metric. The asymptotic distri- bution is calculated and shown to be tightly connected to Minkowski’s question mark function. We proceed to show that the limit distribution is stationary with respect to a random walk on G defined by a specific measure µ. We further prove a stronger result stating that the asymptotic distribution is the limit point for any probability measure over X pushed forward by the convolution power µ∗n.