Table of contents, Bernoulli Journal, vol. 2, no. 3 (1996)

Inference in hidden Markov models I: Local asymptotic normality in the stationary case; Authors: Peter J. Bickel, Ya'acov Ritov

L^p adaptive density estimation; Authors: Gérard Kerkyacharian, Dominique Picard, Karine Tribouley

Variance of Lipschitz functions and an isoperimetric problem for a class of product measures; Authors: Sergei G. Bobkov, Christian Houdré

Almost sure oscillation of certain random processes; Authors: Jean-Marc Azaïs, Mario Wschebor

Laplace and the origin of the Ornstein-Uhlenbeck process; Authors: Martin Jacobsen

On the stochastic equation L(X)=L[B(X+C)] and a property of gamma distributions; Authors: Daniel Dufresne

Following up on work by Baum and Petrie published 30 years ago, we study likelihood-based methods in hidden Markov models, where the hiding mechanism can lead to continuous observations and is itself governed by a parametric model. We show that procedures essentially equivalent to maximum likelihood estimates are asymptotically normal as expected and consistent estimates of the variance can be constructed, so that the usual inferential procedures are asymptotically valid.

We provide global adaptive wavelet-type density estimates. Our procedures illustrate the refinement which can be obtained by replacing the Fourier basis by the wavelet basis in estimation methods. The main argument consists in observing that the estimated total energy of the details of a specified level j will be smaller or greater than some known threshold if precisely j is above or below the theoretical optimal level calculated with the a priori knowledge of the regularity of the density. This balancing effect leads directly to an adaptation procedure, and some natural extensions. We investigate the minimax properties of these procedures and explain their evolution for different global error measures.

The maximal variance of Lipschitz functions (with respect to the ℓ¹-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.

We show that for various classes of stochastic process, namely Gaussian processes, stable Lévy processes and Brownian martingales, we have almost sure weak convergence of the oscillation in the measure space ([0,1],λ), λ being Lebesgue measure. This result is used to obtain almost sure weak approximation of the occupation measure via numbers of crossings.

It is shown that a second-order partial differential equation, found by Laplace in 1810, is the Fokker-Planck equation for a one-dimensional Ornstein-Uhlenbeck process. It is argued that Laplace's reasoning, although not rigorous, can be entirely justified by using the modern theory of weak convergence of stochastic processes. The solutions to the differential equation found by Laplace and others, using expansions in terms of Hermite polynomials, are discussed.

This paper is concerned with the stochastic equation X = B(X+C), where B, X and C are independent. This equation has appeared in a number of contexts, notably in actuarial science. An apparently new property of gamma variables (Theorem 1) leads to the derivation of a new explicit example of solution of the stochastic equation (Theorem 2), where B is the product of two independent beta variables, C is gamma and X is the product of independent beta and gamma variables. Also, a number of previously known explicit examples are seen to be direct algebraic consequences of a well-known property of gamma variables.