On estimation of the L r norm of a regression function

f be observed with noise. In the present paper we study the problem of nonparametric estimation of certain nonsmooth functionals of f, specifically, L _r norms ||f|| _r of f. Known from the literature results on functional estimation deal mostly with two extreme cases: estimating a smooth (differentiable in L ₂ ) functional or estimating a singular functional like the value of f at certain point or the maximum of f. In the first case, the convergence rate typically is n ^−1/2, n being the number of observations. In the second case, the rate of convergence coincides with the one of estimating the function f itself in the corresponding norm. We show that the case of estimating ||f|| _r is in some sense intermediate between the above extremes. The optimal rate of convergence is worse than n ^−1/2 but is better than the rate of convergence of nonparametric estimates of f. The results depend on the value of r. For r even integer, the rate occurs to be n ^{−β/(2β+1−1/r)} where β is the degree of smoothness. If r is not an even integer, then the nonparametric rate n ^{−β/(2β+1)} can be improved, but only by a logarithmic in n factor. Received: 6 February 1996hinspaceairsp/Revised version: 10 June 1998     

Asymptotic equivalence for nonparametric generalized linear models

Summary. We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f(t _i ) of a regression function f at a grid point t _i (nonparametric GLM). When f is in a H?lder ball with exponent we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process. Received: 4 February 1997     

Malliavin calculus and asymptotic expansion for martingales

Summary. We present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale. The emphasis is put on the use of the Malliavin calculus; the uniform nondegeneracy of the Malliavin covariance under certain truncation plays an essential role as the Cramér condition did in the case of independent observations. Applications to statistics are presented. Received: 5 September 1995 / In revised form: 20 October 1996     

Finiteness properties for subgroups of GL (n,\mathbb Z)(n,\mathbb Z)

Abstract. We construct finitely presented subgroups of GL that have infinitely many conjugacy classes of finite subgroups. This answers a question of Grunewald and Platonov. We suggest a variation on their question. Received: 26 August 1999 / Revised: 28 September 1999 / Published online: 8 May 2000     

On the conditioned exit measures of super Brownian motion

In this paper we present a martingale related to the exit measures of super Brownian motion. By changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain. The results are similar in flavor to the “immortal particle” picture of conditioned super Brownian motion but more general, as the change of measure is given by a martingale which need not arise from a single harmonic function. Received: 27 August 1998 / Revised version: 8 January 1999     

Multiple levels of symmetry breaking

In the previous paper in this volume we have studied the p-spin interaction model just below the critical temperature, and we have rigorously proved several aspects of the physicists prediction that this model exhibits “one level of symmetry breaking”. In the present paper we show how to construct systems that exhibit an arbitrarily large, but finite number of “levels of symmetry-breaking”. As the temperature decreases, such systems exhibit many phase transitions, as the structure of the overlaps gains complexity. This phenomenon does not seem to have been described previously, even in the physics literature. Received: 15 January 1998 / Revised version: 10 November 1999 / Published online: 21 June 2000     

Cone paths for the planar Brownian snake

Summary. For the Brownian path-valued process of Le Gall (or Brownian snake) in , the times at which the process is a cone path are considered as a function of the size of the cone and the terminal position of the path. The results show that the paths for the path-valued process have local properties unlike those of a standard Brownian motion. Received: 29 January 1996 / In revised form: 21 June 1996     

Exact capacity results for stable processes

Exact results are proved for the capacity of pullbacks of analytic sets by stable processes. Received: 25 May 1988 / Revised version: 15 September 1997     

A super-Brownian motion with a locally infinite catalytic mass

Summary. A super-Brownian motion in with “hyperbolic” branching rate , is constructed, which symbolically could be described by the formal stochastic equation (with a space-time white noise ). Starting at this superprocess will never hit the catalytic center: There is an increasing sequence of Brownian stopping times strictly smaller than the hitting time of such that with probability one Dynkin's stopped measures vanish except for finitely many Received: 27 November 1995 / In revised form: 24 July 1996     

The Sherrington–Kirkpatrick model: a challenge for mathematicians

Summary. The Sherrington–Kirkpatrick (SK) model for spin glasses is deceptively simple to state. Yet its rigorous study represents a considerable challenge. We report here some modest progresses (obtained through elementary methods). Even in the supposedly simple high temperature region, a number of basic questions remain unsolved. Received: 7 December 1995 / In revised form: 6 March 1997     

Asymptotic summation of power series

Summary. We give an asymptotic expansion in powers of of the remainder , when the sequence has a similar expansion. Contrary to previous results, explicit formulas for the computation of the coefficients are presented. In the case of numerical series (), rigorous error estimates for the asymptotic approximations are also provided. We apply our results to the evaluation of , which generalizes various summation problems appeared in the recent literature on convergence acceleration of numerical and power series. Received April 22, 1997     

Asymptotic behaviour of disconnection and non-intersection exponents

Summary. We study the asymptotic behaviour of disconnection and non-intersection exponents for planar Brownian motionwhen the number of considered paths tends to infinity. In particular, if η _n (respectively ξ (n, p)) denotes the disconnection exponent for n paths (respectively the non-intersection exponent for n paths versus p paths), then we show that lim _{n →∞} η _n /n = 1 2 and that for a > 0 and b > 0,lim _{n →∞} ξ ([na],[nb])/n = (√ a + √ b) ² /2. Received: 28 February 1996 / In revised form: 3 September 1996     

Homogeneous nilmanifolds attached to representations of compact Lie groups

For each compact Lie algebra ? and each real representation V of ? we consider a two-step nilpotent Lie group N(?,V), endowed with a natural left-invariant riemannian metric. The homogeneous nilmanifolds so obtained are precisely those which are naturally reductive. We study some geometric aspects of these manifolds, finding many parallels with H-type groups. We also obtain, within the class of manifolds N(?,V), the first examples of non-weakly symmetric, naturally reductive spaces and new examples of non-commutative naturally reductive spaces. Received: 16 September 1998 / Revised version: 24 February 1999     

Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion

Any solution of the functional equation

Rigorous results for the Hopfield model with many patterns

Summary. We perform a thorough investigation of the main aspects of the Hopfield model with many patterns. Advances are made toward the validity of the “replica symmetric” solution. Strong evidence of the validity of this solution is given over the entire domain where this validity is conjectured; complete proof is given in a subregion that contains strictly the ergodic region. Received: 22 May 1996 / In revised form: 20 May 1997     

Un théorème central limite presque sûr à moments généralisés pour les rotations irrationnelles

In a recent work, we indicated another formulation of the Almost Sure Central Limit Theorem (A.S.C.L.T.), with series in place of averages, by showing that the property of the A.S.C.L.T. directly follows from the theory of orthogonal sums. For, we used the notion of quasi-orthogonal systems introduced earlier by R. Bellmann, and later developed by Kac–Salem–Zygmund. The main object of this paper is to prove a similar result for irrational rotations of the torus. We prove the existence of a generalized moment version of the A.S.C.L.T., with a speed of convergence. In our strategy, we use again the notion of quasi-orthogonal system, and purpose a Gaussian randomization technic, new at least in this context. The proof avoid notably the use of Volny's result on the existence of good Gaussian approximations in aperiodic dynamical systems, and should also permit to be able to treat problems of comparable nature, in particular in non-ergodic cases. Received: 2 February 1999