Generalized level crossings and tangencies of a random field with smooth sample functions

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Tangencies and level crossings of a random field X : R m + × Ω → R n (which is not necessarily Gaussian) are studied under the assumption that almost every sample path is continuously differentiable. If n = m and if the random field has uniformly bounded sample derivatives and uniformly bounded densities for the distributions of the X l , then for a compact K ⊂ R m + and any fixed level, the restriction to K of almost every sample path has no tangencies to the level and at most finitely many crossings. The case of n ≠ m is also examined. Some generic properties, which hold for a residual set of random fields, are analyzed. Proofs involve the concepts of regularity and transversality from differential topology.