Abstract. A random variable Z satisfying the random variable dilation equation MZ d = Z + G, where G is a discrete random variable independent of Z with values in a lattice Γ ⊂ R d and weights {c k } k∈Γ and M is an expanding and Γ-preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density ϕ which will satisfy a dilation equationWe have obtained necessary and sufficient conditions for the existence of the density ϕ and a simple sufficient condition for ϕ's existence in terms of the weights {c k } k∈Γ . Wavelets in R d can be generated in several ways. One is through a multiresolution analysis of L 2 R d generated by a compactly supported prescale function ϕ. The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of ϕ allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when ϕ is a prescale function.

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