Hausdorff dimension of the graph of the fractional Brownian sheet. (English) Zbl 1057.60033
Let \(\{B^{(\alpha)}(t)\}_{t\in R^d}\) be the fractional Brownian sheet with multi-index \(\alpha=(\alpha_1,\dots,\alpha_d)\), \(0<\alpha_i<1\). Kamont has shown that, with probability \(1\), the box dimension of the graph of a trajectory of this Gaussian field, over a non-degenerate cube \(Q\subset R^d\), is equal to \(d+1-\min(\alpha_1,\dots,\alpha_d)\). This paper proves that this result remains true when the box dimension is replaced by the Hausdorff dimension or the packing dimension.
Reviewer: Yuhu Feng (Shanghai)
MSC:
| 60G15 | Gaussian processes |
| 60G17 | Sample path properties |
| 60G60 | Random fields |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 60G50 | Sums of independent random variables; random walks |
Keywords:
Gaussian fields; fractional Brownian motion; random wavelet series; Hausdorff dimension; packing dimensionReferences:
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