In this talk, we consider a special case of infinite urn schemes first considered by Karlin (1967), especially its occupancy and odd-occupancy processes. We propose a natural randomization of these two processes and investigate their decompositions. We then establish functional central limit theorems, showing that each randomized process and its components converge jointly to a decomposition of certain self-similar Gaussian process. In particular, the randomized occupancy process and its components converge jointly to the decomposition of a time-changed Brownian motion $B(t^\alpha)$, $\alpha\in(0, 1)$, and the randomized odd-occupancy process and its components converge jointly to a decomposition of fractional Brownian motion with Hurst index $H\in(0,1/2)$.