A non-uniform Berry–Esseen bound via Stein's method

This paper is part of our efforts to develop Stein's method beyond uniform bounds in normal approximation. Our main result is a proof for a non-uniform Berry–Esseen bound for independent and not necessarily identically distributed random variables without assuming the existence of third moments. It is proved by combining truncation with Stein's method and by taking the concentration inequality approach, improved and adapted for non-uniform bounds. To illustrate the technique, we give a proof for a uniform Berry–Esseen bound without assuming the existence of third moments. Received: 2 March 2000 / Revised version: 20 July 2000 / Published online: 26 April 2001     

Sharp probability estimates for generalized Smirnov statistics

We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0,1] random variables stays to one side of a given line. Author’s address: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA     

Stochastic functional differential equations on manifolds

In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters. Received: 6 July 1999 / Revised version: 19 April 2000 /?Published online: 14 June 2001     

Hölder continuity for spatial and path processes via spectral analysis

For ν(dθ), a σ-finite Borel measure on R ^d , we consider L ²(ν(dθ))-valued stochastic processes Y(t) with te property that Y(t)=y(t,·) where y(t,θ)=∫ ^t ₀ e ^{−λ(θ)( t − s )} dm(s,θ) and m(t,θ) is a continuous martingale with quadratic variation [m](t)=∫ ^t ₀ g(s,θ)ds. We prove timewise H?lder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions of the form dX=AXdt+GdW with W a cylindrical Brownian motion. Maximal inequalities and H?lder continuity results are also provenfor the path process _t (τ)≗Y(τt∧t). Received: 25 June 1999 / Revised version: 28 August 2000 /?Published online: 9 March 2001     

Rigorous low-temperature results for the mean field p-spins interaction model

We prove that, just below the critical temperature, the mean field p-spins interaction model, for p suitably large, spontaneously decomposes into different states. The asymptotic overlaps between any two different states are zero. Under a mild (unproven) hypothesis on the weight distribution of these states, we prove that they are pure states. This situation is called in physics “one level of symmetry breaking”. Received: 15 January 1998 / Revised version: 10 November 1999 / Published online: 21 June 2000     

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     

The high temperature case for the random K-sat problem

We give a completely rigorous proof that the replica-symmetric solution holds at high enough temperature for the random K-sat problem. The most notable feature of this problem is that the order parameter of the system is a function and not a number. Received: 21 April 1998 / Revised version: 24 April 2000 / Published online: 21 December 2000     

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     

Tail probability approximations for Student's t-statistics

In this paper, we derive saddlepoint approximations for Student's t-statistics for strongly nonlattice random variables without moment conditions. Under very mild conditions, we show that saddlepoint equations always have solutions. Supported in part by the grants R-155-050-055-133/101 and R-155-000-035-112 at the National University of Singapore     

On the genus of a maximal curve

The upper limit and the first gap in the spectrum of genera of -maximal curves are known, see [34], [16], [35]. In this paper we determine the second gap. Both the first and second gaps are approximately constant times , but this does not hold true for the third gap which is just 1 for while (at most) constant times q for This suggests that the problem of determining the third gap which is the object of current work on -maximal curves could be intricate. Here, we investigate a relevant related problem namely that of characterising those -maximal curves whose genus is equal to the third (or possible the forth) largest value in the spectrum. Our results also provide some new evidence on -maximal curves in connection with Castelnuovo's genus bound, Halphen's theorem, and extremal curves. Received: 1 January 2001 / Revised version: 30 July 2001 / Published online: 23 May 2002     

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     

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     

On a mapping approach to investigating the bootstrap accuracy

Summary. A simple mapping approach is proposed to study the bootstrap accuracy in a rather general setting. It is demonstrated that the bootstrap accuracy can be obtained through this method for a broad class of statistics to which the commonly used Edgeworth expansion approach may not be successfully applied. We then consider some examples to illustrate how this approach may be used to find the bootstrap accuracy and show the advantage of the bootstrap approximation over the Gaussian approximation. For the multivariate Kolmogorov–Smirnov statistic, we show the error of bootstrap approximation is as small as that of the Gaussian approximation. For the multivariate kernel type density estimate, we obtain an order of the bootstrap error which is smaller than the order of the error of the Gaussian approximation given in Rio (1994). We also consider an application of the bootstrap accuracy for empirical process to that for the copula process. Received: 23 June 1995 / In revised form: 18 June 1996     

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     

On the regularity of the distribution of the maximum of one-parameter Gaussian processes

The main result in this paper states that if a one-parameter Gaussian process has C ^2k paths and satisfies a non-degeneracy condition, then the distribution of its maximum on a compact interval is of class C ^k . The methods leading to this theorem permit also to give bounds on the successive derivatives of the distribution of the maximum and to study their asymptotic behaviour as the level tends to infinity. Received: 14 May 1999 / Revised version: 18 October 1999 / Published online: 14 December 2000     

Order of magnitude bounds for expectations of Δ2-functions of generalized random bilinear forms

Let Φ be a symmetric function, nondecreasing on [0,∞) and satisfying a Δ₂ growth condition, (X ₁,Y ₁), (X ₂,Y ₂),…,(X _n ,Y _n ) be arbitrary independent random vectors such that for any given i either Y _i =X _i or Y _i is independent of all the other variates. The purpose of this paper is to develop an approximation of

The almost equivalence of pairwise and mutual independence and the duality with exchangeability

For a large collection of random variables in an ideal setting, pairwise independence is shown to be almost equivalent to mutual independence. An asymptotic interpretation of this fact shows the equivalence of asymptotic pairwise independence and asymptotic mutual independence for a triangular array (or a sequence) of random variables. Similar equivalence is also presented for uncorrelatedness and orthogonality as well as for the constancy of joint moment functions and exchangeability. General unification of multiplicative properties for random variables are obtained. The duality between independence and exchangeability is established through the random variables and sample functions in a process. Implications in other areas are also discussed, which include a justification for the use of mutually independent random variables derived from sequential draws where the underlying population only satisfies a version of weak dependence. Macroscopic stability of some mass phenomena in economics is also characterized via almost mutual independence. It is also pointed out that the unit interval can be used to index random variables in the ideal setting, provided that it is endowed together with some sample space a suitable larger measure structure. Received: 16 April 1997 / Revised version: 18 May 1998     

On the first Hochschild cohomology group of an algebra

Let A≡KΔ /I be a factor of a path algebra. We develop a strategy to compute dim H ¹(A), the dimension of the first Hochschild cohomology group of A, using combinatorial data from (Δ,I). That allows us to connect dim H ¹(A) with the rank and p-rank of the fundamental group π₁(Δ,I) of (Δ,I). We get explicit formulae for dim H ¹(A), when every path in Δ parallel to an arrow belongs to I or when I is homogeneous. Received: 12 April 1999 / Revised version: 9 October 2000