Table of contents, Bernoulli Journal, vol. 3, no. 1 (1997)

A central limit theorem for normalized functions of the increments of a diffusion process, in the presence of round-off errors; Authors: Sylvain Delattre, Jean Jacod

Information geometry of estimating functions in semi-parametric statistical models; Authors: Shun-Ichi Amari, Motoaki Kawanabe

Efficient estimation in the bivariate normal copula model: normal margins are least favourable; Authors: Chris A.J. Klaassen, Jon A. Wellner

Partition structures derived from Brownian motion and stable subordinators; Authors: Jim Pitman

Recurrence and transience of diffusions in a half-space; Authors: Srinivasan Balaji, Sundareswaran Ramasubramanian

Let $X$ be a one-dimensional diffusion process. For each $n\ge 1$ we have a round-off level ${\alpha }_n>0$ and we consider the rounded-off value $X_t^{\left(\alpha \right)}=\alpha [X_t/\alpha ]$. We are interested in the asymptotic behaviour of the processes $U(n,\phi {)}_t=\frac{1}{n}{\sum }_{1\le i\le \left[\mathrm{nt}\right]}\phi (X_{(i-1)/n}^{\left(\alpha \right)},\sqrt{n}(X_{i/n}^{\left(\alpha \right)}-X_{(i-1)/n}^{\left(\alpha \right)}))$ as $n$ goes to $+\mathrm{\infty }$: under suitable assumptions on $\phi$, and when the sequence $\alpha \sqrt{n}$ goes to a limit $\beta \in [0,\mathrm{\infty })$, we prove the convergence of $U(n,\phi )$ to a limiting process in probability (for the local uniform topology), and an associated central limit theorem. This is motivated mainly by statistical problems in which one wishes to estimate a parameter occurring in the diffusion coefficient, when the diffusion process is observed at times $i/n$ and is subject to rounding off at some level ${\alpha }_n$ which is 'small' but not 'very small'.

For semi-parametric statistical estimation, when an estimating function exists, it often provides an efficient or a good consistent estimator of the parameter of interest against nuisance parameters of infinite dimensions. The present paper elucidates the structure of estimating functions, based on the dual differential geometry of statistical inference and its extension to fibre bundles. The paper studies the following problems. First, when does an estimating function exist and what is the set of all the estimating functions? Second, how are the asymptotic variances of the estimators derived from estimating functions and when are the estimators efficient? Third, how do we adaptively choose a practically good (quasi-)estimating function from the observed data? The concept of m-curvature freeness plays a fundamental role in solving the above problems.

Consider semi-parametric bivariate copula models in which the family of copula functions is parametrized by a Euclidean parameter θ of interest and in which the two unknown marginal distributios are the (infinite-dimensional) nuisance parameters. The efficient score for θ can be characterized in terms of the solutions of two coupled Sturm-Liouville equations. Where the family of copula functions corresponds to the normal distributios with mean 0, variance 1 and correlation θ, the solution of these equations is given, and we thereby show that the normal scores rank correlation coefficient is asymptotically efficient. We also show that the bivariate normal model with equal variances constitutes the least favourable parametric submodel. Finally, we discuss the interpretation of |θ| in the normal copula model as the maximum (monotone) correlation coefficient.

Explicit formulae are obtained for the distribution of various random partitions of a positive integer n, both ordered and unordered, derived from the zero set M of a Brownian motion by the following scheme: pick n points uniformly at random from [0,1], and classify them by whether they fall in the same or different component intervals of the complement of M. Corresponding results are obtained for M the range of a stable subordinator and for bridges defined by conditioning on 1∈M. These formulae are related to discrete renewal theory by a general method of discretizing a subordinator using the points of an independent homogeneous Poisson process.

For non-degenerate diffusions in the half-space with oblique reflection, a dichotomy between recurrence and transience is established; convenient characterizations of recurrence and transience are given. Verifiable criteria for recurrence/transience are derived in terms of the generator and the boundary operator. Using these criteria, `real variables proofs' of some results due to Rogers, concerning reflecting Brownian motion in a half-plane, are obtained. The problem of transience down a side in the case of diffusions in the half-plane is dealt with. Positive recurrence of diffusions in half-space is also considered; it is shown that the hitting time of any open set has finite expectation if there is just one positive recurrent point.

Orbituary Boris Vladimirovitch Gnedenko (1912 - 1995)