Moments of discrete measures with dense jumps induced by -expansions

Let β > 1 . Through an appeal to β -expansions we define a strictly increasing and left-continuous function μ β on [ 0 , 1 ] . Then μ β turns out to be a pure jump distribution. In other words, its associated Lebesgue–Stieltjes measure is discrete, i.e., a summation of point masses. The present note studies the moment of this discrete measure and its asymptotics.