Let $P$ be a probability measure defined on a $\sigma$-field $\mathscr{F}$ over $\Omega$. Let $\mathfrak{L}\subset \mathscr{F}$ be a $\sigma$-lattice and $r > 1$. For each $A \in \mathscr{F}$ denote by $P_r(A/\mathfrak{L})$ the unique nearest point projection of $1_A$ onto the closed convex subspace of all "$\mathfrak{L}$-measurable" equivalence-classes of $L_r(\Omega, \mathscr{F}, P)$. It is shown that there exists a functional relationship between $P_r(A/\mathfrak{L})$ and $P_2(A/\mathfrak{L})$ of the form $$P_r(A/\mathfrak{L}) = \varphi(P_2(A/\mathfrak{L}))$$ where the function $\varphi$ depends only on $r$ but not on $A, P$ or $\mathfrak{L}$. This relationship is applied to the theory of sufficiency.
Publié le : 1979-02-14
Classification:
Projection in $L_r$,
conditional expectation,
$\sigma$-lattice,
sufficiency,
46E30,
62B05
@article{1176995159,
author = {Landers, D. and Rogge, L.},
title = {A Functional Relationship between the Different $r$-means for Indicator Functions},
journal = {Ann. Probab.},
volume = {7},
number = {6},
year = {1979},
pages = { 166-169},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995159}
}
Landers, D.; Rogge, L. A Functional Relationship between the Different $r$-means for Indicator Functions. Ann. Probab., Tome 7 (1979) no. 6, pp. 166-169. http://gdmltest.u-ga.fr/item/1176995159/