## Bernoulli

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

### 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.\$

#### Article information

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

Dates
First available in Project Euclid: 4 March 2004

https://projecteuclid.org/euclid.bj/1078435223

Mathematical Reviews number (MathSciNet)
MR2003m:60131

Zentralblatt MATH identifier
1012.60074

#### Citation

Kühn, Thomas; Linde, Werner. Optimal series representation of fractional Brownian sheets. Bernoulli 8 (2002), no. 5, 669--696. https://projecteuclid.org/euclid.bj/1078435223

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