(Georgia Institute of Technology, 1995-09-16)
Bobkov, S. G.; Houdré, Christian
The optimal constants in the multiplicative Sobolev inequalities where the gradient is estimated in the L_1-norm and the function in two different Lebesgue norms are found. With the optimal constants, these inequalities turn out to still be equivalent to the isoperimetric
property of the balls in the Euclidean space. In the course of the proof, relations between Lorentz and Lebesgue spaces are studied (and also
applied to some different measures, e.g., Riesz potentials).
(Georgia Institute of Technology, 1995-08)
Cheng, R.; Houdré, Christian
This work is concerned with the prediction problem for a class of
L^p-random fields. For this class of fields, we derive prediction error
formulas, spectral factorizations, and orthogonal decompositions.
(Georgia Institute of Technology, 1995-07-25)
Bobkov, S. G.; Houdré, Christian
If the half-spaces of the form {x\in R^n: x_1 \le c} are extremal in
the isoperimetric problem for the product measure \mu^n, n\ge 2, then
\mu is Gaussian.
(Georgia Institute of Technology, 1995-07-24)
Bobkov, S. G.; Houdré, Christian
For product probability measures \mu^n, we obtain necessary and sufficient conditions (in terms of \mu) for dimension free isoperimetric inequalities of the form
\mu^n (A + h[-1,1]^n)\ge R_h(\mu^n(A)) to hold; for a function R such that R(p) > p, for
all (some) p \in (0,1), and for h > 0 large enough. Some questions related to the
concentration of measure phenomenon are also discussed.
(Georgia Institute of Technology, 1995-07-10)
Bobkov, S. G.; Houdré, Christian
The maximal variance of Lipschitz functions (with respect to the
\ell_1-distance) of independent random vectors is found. This is then used
to solve the isoperimetric problem, uniformly in the class of product
probability measures with given variance.
(Georgia Institute of Technology, 1995-04)
Houdré, Christian; Lacey, Michael T.
It is shown here how to extend the spectral characterization of the
strong law of large numbers for weakly stationary processes to certain singular
averages. For instance, letting {X_t, t \in R^3}, be a weakly stationary field, {X_t}
satisfies the usual SLLN (by averaging over balls) if and only if the averages of
{X_t} over spheres of increasing radii converge pointwise. The same result in two
dimensions is false. This spectral approach also provide a necessary and sufficient
condition for the a.s. convergence of some series of stationary variables.