• Bernoulli
  • Volume 8, Number 5 (2002), 669-696.

Optimal series representation of fractional Brownian sheets

Thomas Kühn and Werner Linde

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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

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


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

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.$

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Bernoulli, Volume 8, Number 5 (2002), 669-696.

First available in Project Euclid: 4 March 2004

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approximation numbers fractional Brownian motion Gaussian process small-ball behaviour


Kühn, Thomas; Linde, Werner. Optimal series representation of fractional Brownian sheets. Bernoulli 8 (2002), no. 5, 669--696.

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