kth records from geometric distribution

Let (Rn(k),n≥0) be a sequence of kth records arising from the geometric distribution. We show that if k≥2 then Rn(k) and Rn+1(k)−Rn(k) are independent or the regression of Rn+1(k)−Rn(k) given Rn(k) is constant iff n=0. Then we present several characterizations of the geometric distribution or its tail based on the properties of kth records.