Propri�t�s de convergence presque compl�te du pr�dicteur � noyau

Abstract

Let (Z n )ℕ be a φ-mixing process which is valued in % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbf9F4rqqrpu0dXde9LqFHe9Lq% pG0di9Veea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqI+frFve9Fve9% Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1uy0H% MmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-ri8fjabgkOimlab-1ri% snaaCaaaleqabaGaamyCaaaaaaa!47BE! $$\mathbb{E} \subset \mathbb{R}^q $$ , g be a real measurable function defined on % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbf9F4rqqrpu0dXde9LqFHe9Lq% pG0di9Veea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqI+frFve9Fve9% Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1uy0H% MmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-ri8fbaa!438A! $$\mathbb{E}$$ , s and k be two positive integers. We suppose the existence of a function R satisfying R(.) = E(g(Z n+s )/[Z n−k+1 ,....Z n ]=.), ∀n∈ℕ, n≧k, and estimate the function R from a sequence {Z i, i=1, ..., n} by R n with % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbf9F4rqqrpu0dXde9LqFHe9Lq% pG0di9Veea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqI+frFve9Fve9% Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbWaaSbaaS% qaaiaad6gaaeqaaOGaaiikaiaadwhacaGGPaGaeyypa0ZaaSaaaeaa% daaeWbqaaiaadEgacaGGOaGaamOwamaaBaaaleaacaWGPbGaey4kaS% Iaam4CaaqabaGccaGGPaGaam4saiaacIcacaGGBbGaamyDaiabgkHi% TiaacIcacaWGAbWaaSbaaSqaaiaadMgacqGHsislcaWGRbGaey4kaS% IaaGymaaqabaGccaGGSaGaeS47IWKaaiilaiaadQfadaWgaaWcbaGa% amyAaaqabaGccaGGPaGaaiyxaiaadIgadaqhaaWcbaGaamOBaaqaai% abgkHiTiaaigdaaaGccaGGPaaaleaacaWGPbGaeyypa0Jaam4Aaaqa% aiaad6gacqGHsislcaWGZbaaniabggHiLdaakeaadaaeWbqaaiaadU% eacaGGOaGaai4waiaadwhacqGHsislcaGGOaGaamOwamaaBaaaleaa% caWGPbGaeyOeI0Iaam4AaiabgUcaRiaaigdaaeqaaOGaaiilaiabl+% UimjaacYcacaWGAbWaaSbaaSqaaiaadMgaaeqaaOGaaiykaiaac2fa% caWGObWaa0baaSqaaiaad6gaaeaacqGHsislcaaIXaaaaOGaaiykaa% WcbaGaamyAaiabg2da9iaadUgaaeaacaWGUbGaeyOeI0Iaam4Caaqd% cqGHris5aaaakiaacYcaieaacaWFGaGaa8hiaiaa-bcacqGHaiIica% WG1bGaeyicI48efv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39ga% iuaacqGFecFrdaahaaWcbeqaaiaadUgaaaaaaa!9197! $$R_n (u) = \frac{{\sum\limits_{i = k}^{n - s} {g(Z_{i + s} )K([u - (Z_{i - k + 1} , \cdots ,Z_i )]h_n^{ - 1} )} }}{{\sum\limits_{i = k}^{n - s} {K([u - (Z_{i - k + 1} , \cdots ,Z_i )]h_n^{ - 1} )} }}, \forall u \in \mathbb{E}^k $$ where K is a kernel of ℝkq and h n∈ℝ, h n>0. We give various conditions, on both the sequences (h n )ℕ and the sequence (φ n )ℕ which is associated with (Z n )ℕ, for the two following properties: uniform complete convergence to R for the functional estimator R n and complete convergence to 0 for the real random variable R n(Z n−k+1, ..., Z n)−R(Zn−k+1, ..., Z n). This last result concerns the predictor R n(Z n−k+1, ..., Z n) of g(Z n+s) from {Z i, i=1, ..., n} when the process (Z n )ℕ is stationary and markovian of order k. These results are proved here for a more general problem: estimation of a regression E(Y/X) from {(X i, Y i), i=1, ..., n} when these couples are not independent. We also give a lemma which is an extension of the Bernstein inequality to the case of φ-mixing r.r.v.