Abstract. For a stationary random field (X j ) j∈Z d and some measure µ on R d , we consider the set-indexed weighted sum process Sn(A) = j∈Z d µ(nA ∩ R j ) 1 2 X j , where R j is the unit cube with lower corner j. We establish a general invariance principle under a p-stability assumption on the X j 's and an entropy condition on the class of sets A. The limit processes are selfsimilar set-indexed Gaussian processes with continuous sample paths. Using Chentsov's type representations to choose appropriate measures µ and particular sets A, we show that these limits can be Lévy (fractional) Brownian fields or (fractional) Brownian sheets.
International audienceWe 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 $\mathbb B(t^\alpha), \alpha\in(0,1)$, and the randomized odd-occupancy process and its components converge jointly to a decomposition of fractional Brownian motion with Hurst index $H\in(0,1/2)$. 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
We present a new technique for proving the empirical process invariance principle for stationary processes (X n ) n≥0 . The main novelty of our approach lies in the fact that we only require the central limit theorem and a moment bound for a restricted class of functions ( f (X n )) n≥0 , not containing the indicator functions. Our approach can be applied to Markov chains and dynamical systems, using spectral properties of the transfer operator. Our proof consists of a novel application of chaining techniques.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.