Abstract
Let X t, t= ..., \s-1,0,1,... be a strietly stationary sequence of random variables (r.v.'s) defined on a probability space (Ω,P) and taking values in R d.Let X 1,...,X nbe n consecutive observations of X t.Let f be the density of X 1.As an estimator of f(x), we shall consider % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaaiaacaqabeaadaqaaqGaaO% qaaiqadAgagaqcamaaBaaaleaacaWGUbaabeaakiaacIcacaWG4bGa% aiykaiabg2da9iaad6gadaahaaWcbeqaaiabgkHiTiaaigdaaaGcda% aeWbqaaiaadkgadaWgaaWcbaGaamOAaaqabaGcdaahaaWcbeqaaiab% gkHiTiaadsgaaaGccaWGlbGaaiikaiaacIcacaWG4bGaeyOeI0Iaam% iwamaaBaaaleaacaWGQbaabeaakiaacMcacaGGVaGaamOyamaaBaaa% leaacaWGQbaabeaakiaacMcaaSqaaiaadQgacqGH9aqpcaaIXaaaba% GaamOBaaqdcqGHris5aaaa!58A9!\[\hat f_n (x) = n^{ - 1} \sum\limits_{j = 1}^n {b_j ^{ - d} K((x - X_j )/b_j )} \]. Here K is a kernel function and b nis a esquence of bandwidths tending to zero as n → ∞. The asymptotic distribution and uniform convergence of f n are obtained under general conditions. Appropriate bandwidths are given explicitly. The process X tis assumed to satisfy a weak dependence condition defined in terms of joint densities. The results are applicable to a large class of time series models.
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Tran, L.T. Recursive kernel density estimators under a weak dependence condition. Ann Inst Stat Math 42, 305–329 (1990). https://doi.org/10.1007/BF00050839
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DOI: https://doi.org/10.1007/BF00050839