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We investigate a special case of infinite urn schemes first considered byKarlin (1967), especially its occupancy and odd-occupancy processes. We firstpropose a natural randomization of these two processes and theirdecompositions. We then establish functional central limit theorems, showingthat each randomized process and its components converge jointly to adecomposition of certain self-similar Gaussian process. In particular, therandomized occupancy process and its components converge jointly to thedecomposition of a time-changed Brownian motion $\mathbb B(t^\alpha),\alpha\in(0,1)$, and the randomized odd-occupancy process and its componentsconverge jointly to a decomposition of fractional Brownian motion with Hurstindex $H\in(0,1/2)$. The decomposition in the latter case is a special case ofthe decompositions of bi-fractional Brownian motions recently investigated byLei and Nualart (2009). The randomized odd-occupancy process can also be viewedas correlated random walks, and in particular as a complement to the modelrecently introduced by Hammond and Sheffield (2013) as discrete analogues offractional Brownian motions.

From infinite urn schemes to decompositions of self-similar Gaussian processes