  We present eigenvalue decay estimates of integral operators associated with
compositional dot-product kernels. The estimates improve on previous ones
established for power series kernels on spheres. This allows us to obtain the
volumes of balls in the corresponding reproducing kernel Hilbert spaces. We
discuss the consequences on statistical estimation with compositional dot
product kernels and highlight interesting trade-offs between the approximation
error and the statistical error depending on the number of compositions and the
smoothness of the kernels.
