Optimal series representation of fractional Brownian sheets

Abstract

For 0 $lt; \gamma 2$, let $B_\gamma^d$ be a $d$-dimensional $\gamma$-fractional Brownian sheet with index set $[0,1]^d$ and let $(\xi_k)k \geq 1$ be an independent sequence of standard normal random variables. We prove the existence of continuous functions uk such that almost surely

$$B_γ^d(t)=\sum\limits_{k=1}^\infty \xi_k u_k(t), \qquad t \in[0,1]^d,$$

and

$$ \left({\mathbb E}\sup_{t\in[0,1]^d}\left|\sum\limits_{k=n}^\infty\xi_k\,u_k(t)\right|^2\right)^{1/2} \approx n^{-\gamma/2}\,(1+log n)^{d(\gamma+1)/2\,-\gamma/2} \;$$

This order is shown to be optimal. We obtain small-ball estimates for $B^\gamma_d$, extending former results in the case $\gamma=1$. Our investigations rest upon basic properties of different kinds of $s$-numbers of operators.$