Invariance principles for self-similar set-indexed random fields

Abstract:

For a stationary random field $(X_j)_{j\in \mathbb {Z}^d}$ and some measure $\mu$ on $\mathbb {R}^d$, we consider the set-indexed weighted sum process \[ S_n(A)=\sum _{j\in \mathbb {Z}^d}\mu (nA\cap R_j)^\frac 12 X_j,\] 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 self-similar set-indexed Gaussian processes with continuous sample paths. Using Chentsov’s type representations to choose appropriate measures $\mu$ and particular sets $A$, we show that these limits can be Lévy (fractional) Brownian fields or (fractional) Brownian sheets.