Consider n points independently sampled from a density p of class $\mathcal{C}^2$ on a smooth compact d-dimensional submanifold $\mathcal{M}$ of $\mathbb{R}^m$, and consider the random walk visiting these points according to a transition kernel K. We study the almost sure uniform convergence of the generator of this process to the diffusive Laplace–Beltrami operator when n tends to infinity, from which we establish the convergence of the random walk to a diffusion process on the manifold. In contrast to known results, our result does not require the kernel K to be continuous, which covers the cases of walks exploring k-nearest neighbor (kNN) and geometric graphs, and convergence rates are given. The distance between the random walk generator and the limiting operator is separated into several terms: a statistical term, related to the law of large numbers, is treated with concentration tools and an approximation term that we control with tools from differential geometry. The case of kNN Laplacians is detailed. The convergence of the stochastic processes having these operators as generators is also studied, by establishing additional tightness results of their distributions on the space of càdlàg functions.

Measure of uncertainty in past lifetime distribution plays an important role in the context of information theory, forensic science and other related fields. In the present work, we propose non-parametric kernel type estimator for generalized past entropy function, which was introduced by Gupta and Nanda [9], under $\alpha$-mixing sample. The resulting estimator is shown to be weak and strong consistent and asymptotically normally distributed under certain regularity conditions. The performance of the estimator is validated through simulation study and a real data set.

We develop an integration by parts technique for point processes, with application to the computation of sensitivities via Monte Carlo simulations in stochastic models with jumps. The method is applied to density estimation with respect to the Lebesgue measure via a modified kernel estimator which is less sensitive to variations of the bandwidth parameter than standard kernel estimators. This applies to random variables whose densities are not analytically known and requires the knowledge of the point process jump times.

Let θ be the mode of a probability density and θn itskernel estimator. In the case θ is nondegenerate, we first specify the weakconvergence rate of the multivariate kernel mode estimator by stating the central limittheorem for θn - θ. Then, we obtain a multivariate law ofthe iterated logarithm for the kernel mode estimator by proving that, with probabilityone, the limit set of the sequence θn - θ suitably normalized is an ellipsoid.We also give a law of the iterated logarithm for the lp norms, p ∈ [1,∞], ofθn - θ. Finally, we consider the case θ is degenerate and give the exactweak and strong convergence rate of θn - θ in the univariate framework.