2015/08/06 by Olivier Durieu, Durieu, Olivier, Yizao Wang +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1508.01506
openalex publication_date 2015/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate a special case of infinite urn schemes first considered by Karlin (1967), especially its occupancy and odd-occupancy processes. We first propose a natural randomization of these two processes and 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 \mathbb B(tα), α∈(0,1), and the randomized odd-occupancy process and its components converge jointly to a decomposition of fractional Brownian motion with Hurst index H∈(0,1/2). The decomposition in the latter case is a special case of the decompositions of bi-fractional Brownian motions recently investigated by Lei and Nualart (2009). The randomized odd-occupancy process can also be viewed as correlated random walks, and in particular as a complement to the model recently introduced by Hammond and Sheffield (2013) as discrete analogues of fractional Brownian motions.