Let fn denote a kernel density estimator of a bounded continuous density f in the real line. Let (t) be a positive continuous function such that f β ∞ <∞. Under natural smoothness conditions, necessary and sufficient conditions for the sequence (t)) (properly centered and normalized) to converge in distribution to the double exponential law are obtained. The proof is based on Gaussian approximation and a (new) limit theorem for weighted sup-norms of a stationary Gaussian process. This extends well known results of Bickel and Rosenblatt to the case of weighted sup-norms, with the sup taken over the whole line. In addition, all other possible limit distributions of the above sequence are identified (subject to some regularity assumptions).

Let v be a vector field in a bounded open set G ⊂ R d . Suppose that v is observed with a random noise at random points Xi, i = 1, . . . , n, that are independent and uniformly distributed in G. The problem is to estimate the integral curve of the differential equationstarting at a given point x(0) = x0 ∈ G and to develop statistical tests for the hypothesis that the integral curve reaches a specified set Γ ⊂ G. We develop an estimation procedure based on a Nadaraya-Watson type kernel regression estimator, show the asymptotic normality of the estimated integral curve and derive differential and integral equations for the mean and covariance function of the limit Gaussian process. This provides a method of tracking not only the integral curve, but also the covariance matrix of its estimate. We also study the asymptotic distribution of the squared minimal distance from the integral curve to a smooth enough surface Γ ⊂ G. Building upon this, we develop testing procedures for the hypothesis that the integral curve reaches Γ. The problems of this nature are of interest in diffusion tensor imaging, a brain imaging technique based on measuring the diffusion tensor at discrete locations in the cerebral white matter, where the diffusion of water molecules is typically anisotropic. The diffusion tensor data is used to estimate the dominant orientations of the

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