New PDF release: Analytical Methods in Probability Theory. Proc. conf.

By Daniel Dugue, E. Lukacs, V. K. Rohatgi

ISBN-10: 3540108238

ISBN-13: 9783540108238

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Extra info for Analytical Methods in Probability Theory. Proc. conf. Oberwolfach, 1980

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1). nσ2θ + σ2e 6. Montrer que E[Z] = σ2e mθ nσ2θ + σ2e et VarZ = σ2θ σ2e . nσ2θ + σ2e ^ n = E [Θ | X] en fonction de la moyenne empirique X¯ n = 7. Exprimer Θ mesures. 3 Espérance conditionnelle 8. Montrer que l’erreur quadratique moyenne est E ^n Θ−Θ 2 σ2θ σ2e = . nσ2θ + σ2e Quel est l’intérêt de faire plusieurs mesures ? ^ n convergent presque sûrement quand n → ∞. Quelle est la limite. 9. Montrer que Xn et Θ On parle d’estimateurs convergents. √ √ ^ n −Θ). Comparer les erreurs quadratiques 10.

1 Tribus construites à partir de variables aléatoires Soit Ω un ensemble muni d’une tribu A c’est-à-dire un sous-ensemble de l’ensemble P(Ω) des parties de Ω – contenant ∅ et Ω – stable par passage au complémentaire – stable par union ou intersection dénombrable. 1 Une application X de Ω muni de la tribu A dans R est une variable aléatoire si pour tout B dans la tribu borélienne de R (plus petite tribu qui contient tous les intervalles) : {ω ∈ Ω, X(ω) ∈ B} = {X ∈ B} ∈ A. Cette propriété est la A-mesurabilité de X.

On obtient, sans difficultés, que : Cov (Zα,β , Bs ) = E B t+s Bs − αE(B2s ) − βE(Bt Bs ). 2 On voit donc qu’il faut savoir calculer, pour un mouvement brownien standard E(Bu Bv ) si u ≤ v. Pour cela remarquons que : E(Bu Bv ) = E(B2u + Bu (Bv − Bu )) = E(B2u ) + E(Bu (Bv − Bu )). Mais, Bu et Bv − Bu sont indépendantes et E(Bu ) = 0 donc : E(Bu (Bv − Bu )) = 0. De plus, Bu suit une loi gaussienne centrée de variance u donc E(B2u ) = u. On voit donc que, si u ≤ v : E(Bu Bv ) = u. Plus généralement, cela implique que pour tout t et pour tout s : E(Bt Bs ) = t ∧ s = inf(s, t).

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Analytical Methods in Probability Theory. Proc. conf. Oberwolfach, 1980 by Daniel Dugue, E. Lukacs, V. K. Rohatgi


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