We derive representations of higher order dual measures of risk in $\mathcal{L}^p$ spaces as suprema of integrals of Average Values at Risk with respect to probability measures on $(0,1]$ (Kusuoka representations). The suprema are taken over convex sets of probability measures. The sets are described by constraints on the dual norms of certain transformations of distribution functions. For $p=2$, we obtain a special description of the set and we relate the measures of risk to the Fano factor in statistics.

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