Limit theorems for bivariate Appell polynomials. Part II: Non-central limit theorems

Summary. Let (X _t ,t∈Z) be a linear sequence with non-Gaussian innovations and a spectral density which varies regularly at low frequencies. This includes situations, known as strong (or long-range) dependence, where the spectral density diverges at the origin. We study quadratic forms of bivariate Appell polynomials of the sequence (X _t ) and provide general conditions for these quadratic forms, adequately normalized, to converge to a non-Gaussian distribution. We consider, in particular, circumstances where strong and weak dependence interact. The limit is expressed in terms of multiple Wiener-It? integrals involving correlated Gaussian measures. Received: 22 August 1996 / In revised form: 30 August 1997     

Some central limit theorems for ℓ∞-valued semimartingales and their applications

Summary. This paper is devoted to the generalization of central limit theorems for empirical processes to several types of ℓ^∞(Ψ)-valued continuous-time stochastic processes t⇝X _t ⁿ =(X _t ^{n ,ψ}|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X _t ^{n ,ψ} is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived. Received: 6 May 1996 / In revised form: 4 February 1997     

Asymptotic expansions in limit theorems for stochastic processes. II

. For a certain class of families of stochastic processes η^ε(t), 0≤t≤T, constructed starting from sums of independent random variables, limit theorems for expectations of functionals F(η^ε[0,T]) are proved of the form

Semi-selfdecomposable distributions and a new class of limit theorems

Self-decomposable distributions are given as limits of normalized sums of independent random variables. We define semi-selfdecomposable distributions as limits of subsequences of normalized sums. More generally, we introduce a way of making a new class of limiting distributions derived from a class of distributions by taking the limits through subsequences of normalized sums, and define the class of semi-selfdecomposable distributions and a decreasing sequence of subclasses of it. We give two kinds of necessary and sufficient conditions for distributions belonging to those classes, one is in terms of the decomposability of random variables and another is in terms of Lévy measures. Received: 1 May 1997 / Revised version: 5 February 1998     

A central limit theorem for stationary random fields

Summary. We prove a central limit theorem for strictly stationary random fields under a projective assumption. Our criterion is similar to projective criteria for stationary sequences derived from Gordin's theorem about approximating martingales. However our approach is completely different, for we establish our result by adapting Lindeberg's method. The criterion that it provides is weaker than martingale-type conditions, and moreover we obtain as a straightforward consequence, central limit theorems for α-mixing or φ-mixing random fields. Received: 19 February 1997 / In revised form: 2 September 1997     

Normal limit theorems for symmetric random matrices

Using the machinery of zonal polynomials, we examine the limiting behavior of random symmetric matrices invariant under conjugation by orthogonal matrices as the dimension tends to infinity. In particular, we give sufficient conditions for the distribution of a fixed submatrix to tend to a normal distribution. We also consider the problem of when the sequence of partial sums of the diagonal elements tends to a Brownian motion. Using these results, we show that if O _n is a uniform random n×n orthogonal matrix, then for any fixed k>0, the sequence of partial sums of the diagonal of O ^k _n tends to a Brownian motion as n→∞. Received: 3 February 1998 / Revised version: 11 June 1998     

Asymptotic expansions in limit theorems for stochastic processes. I

Summary. For some families of locally infinitely divisible Markov processes η ^ɛ (t), 0≦ t≦ T, with frequent small jumps, limit theorems for expectations of functionals F(η ^ɛ [0,T]) are proved of the form | E ^ɛ F(η ^ɛ [0,T])−E ⁰ F(η ⁰ [0,T])|≦ const ⋅ k(ɛ) , E ^ɛ F(η ^ɛ [0,T])=E ⁰ [F(η ⁰ [0,T])+ k(ɛ) ⋅ A ₁ F(η ⁰ [0,T])]+o(k(ɛ))  (ɛ↓ 0) , where A ₁ is a linear differential operator acting on functionals, and the constant is expressed in terms of the local characteristics of the processes η ^ɛ (t) and the norms of the derivatives of the functional F. Received: 1 April 1994 / In revised form: 30 September 1995     

One-sided local large deviation and renewal theorems in the case of infinite mean

Summary. If {S _n ,n≧0} is an integer-valued random walk such that S _n /a _n converges in distribution to a stable law of index α∈ (0,1) as n→∞, then Gnedenko’s local limit theorem provides a useful estimate for P{S _n =r} for values of r such that r/a _n is bounded. The main point of this paper is to show that, under certain circumstances, there is another estimate which is valid when r/a _n → +∞, in other words to establish a large deviation local limit theorem. We also give an asymptotic bound for P{S _n =r} which is valid under weaker assumptions. This last result is then used in establishing some local versions of generalized renewal theorems. Received: 9 August 1995 / In revised form: 29 September 1996     

A postprocessed Galerkin method with Chebyshev or Legendre polynomials

Summary. We present an approximate-inertial-manifold-based postprocess to enhance Chebyshev or Legendre spectral Galerkin methods. We prove that the postprocess improves the order of convergence of the Galerkin solution, yielding the same accuracy as the nonlinear Galerkin method. Numerical experiments show that the new method is computationally more efficient than Galerkin and nonlinear Galerkin methods. New approximation results for Chebyshev polynomials are presented. Received January 5, 1998 / Revised version received September 7, 1999 / Published online June 8, 2000     

The central limit theorem for random perturbations of rotations

Summary.   We prove a functional central limit theorem for stationary random sequences given by the transformations

On the estimation of parameters for linear stochastic differential equations

The estimation of arbitrary number of parameters in linear stochastic differential equation (SDE) is investigated. The local asymptotic normality (LAN) of families of distributions corresponding to this SDE is established and the asymptotic efficiency of the maximum likelihood estimator (MLE) is obtained for the wide class of loss functions with polynomial majorants. As an example a single-degree of freedom mechanical system is considered. The results generalize [8], where all elements of the drift matrix are estimated and the asymptotic efficiency is proved only for the bounded loss functions. Received: 12 March 1997 / Revised version: 22 June 1998     

Some large deviation results for sparse random graphs

Summary. We obtain a large deviation principle (LDP) for the relative size of the largest connected component in a random graph with small edge probability. The rate function, which is not convex in general, is determined explicitly using a new technique. The proof yields an asymptotic formula for the probability that the random graph is connected. We also present an LDP and related result for the number of isolated vertices. Here we make use of a simple but apparently unknown characterisation, which is obtained by embedding the random graph in a random directed graph. The results demonstrate that, at this scaling, the properties `connected' and `contains no isolated vertices' are not asymptotically equivalent. (At the threshold probability they are asymptotically equivalent.) Received: 14 November 1996 / In revised form: 15 August 1997     

On signed normal-Poisson approximations

For lattice distributions a convolution of two signed Poisson measures proves to be an approximation comparable with the normal law. It enables to get rid of cumbersome summands in asymptotic expansions and to obtain estimates for all Borel sets. Asymptotics can be constructed two-ways: by adding summands to the leading term or by adding summands in its exponent. The choice of approximations is confirmed by the Ibragimov-type necessary and sufficient conditions. Received: 20 November 1996 / Revised version: 5 December 1997     

A class of bivariate orthogonal polynomials and cubature formula

Summary. The existence of Gaussian cubature for a given measure depends on whether the corresponding multivariate orthogonal polynomials have enough common zeros. We examine a class of orthogonal polynomials of two variables generated from that of one variable. Received February 9, 1993 / Revised version received January 18, 1994     

Inequalities for absolutely regular sequences: application to density estimation

This paper investigates the problem of density estimation for absolutely regular observations. In a first part, we state two important results: a new variance inequality and a Rosenthal type inequality. This allows us to study the ? _p -integrated risk, p≧ 2, of a large class of density estimators including kernel or projection estimators. Under the summability condition on the mixing coefficients ∑ _{k≧ 0} (k+1) ^{p− 2} β _k <∞, the rates obtained are those known to be optimal in the independent setting. Received: 17 October 1995 / In revised form: 26 October 1996     

Fluctuations of empirical means at low temperature for finite Markov chains with rare transitions in the general case

Summary. The integrated autocovariance and autocorrelation time are essential tools to understand the dynamical behavior of a Markov chain. We study here these two objects for Markov chains with rare transitions with no reversibility assumption. We give upper bounds for the autocovariance and the integrated autocorrelation time, as well as exponential equivalents at low temperature. We also link their slowest modes with the underline energy landscape under mild assumptions. Our proofs will be based on large deviation estimates coming from the theory of Wentzell and Freidlin and others [4, 3, 12], and on coupling arguments (see [6] for a review on the coupling method). Received 5 August 1996 / In revised form: 6 August 1997     

The central limit theorem for Markov chains with normal transition operators, started at a point

The central limit theorem and the invariance principle, proved by Kipnis and Varadhan for reversible stationary ergodic Markov chains with respect to the stationary law, are established with respect to the law of the chain started at a fixed point, almost surely, under a slight reinforcing of their spectral assumption. The result is valid also for stationary ergodic chains whose transition operator is normal. Received: 28 March 2000 / Revised version: 25 July 2000 /?Published online: 15 February 2001     

A central limit theorem for “critical” first-passage percolation in two dimensions

Summary. Consider (independent) first-passage percolation on the edges of ℤ ² . Denote the passage time of the edge e in ℤ ² by t(e), and assume that P{t(e) = 0} = 1/2, P{0<t(e)<C ₀ } = 0 for some constant C ₀ >0 and that E[t ^δ (e)]<∞ for some δ>4. Denote by b _0,n the passage time from 0 to the halfplane {(x,y): x ≧ n}, and by T( 0 ,nu) the passage time from 0 to the nearest lattice point to nu, for u a unit vector. We prove that there exist constants 0<C ₁ , C ₂ <∞ and γ _n such that C ₁ ( log n) ^1/2 ≦γ _n ≦ C ₂ ( log n) ^1/2 and such that γ _n ⁻¹ [b _0,n −Eb _0,n ] and (√ 2γ _n ) ⁻¹ [T( 0 ,nu) − ET( 0 ,nu)] converge in distribution to a standard normal variable (as n →∞, u fixed). A similar result holds for the site version of first-passage percolation on ℤ ² , when the common distribution of the passage times {t(v)} of the vertices satisfies P{t(v) = 0} = 1−P{t(v) ≧ C ₀ } = p _c (ℤ ² , site ) := critical probability of site percolation on ℤ ² , and E[t ^δ (u)]<∞ for some δ>4. Received: 6 February 1996 / In revised form: 17 July 1996     

The characterizations of a semicircle law by the certain freeness in a C*-probability space

In usual probability theory, various characterizations of the Gaussian law have been obtained. For instance, independence of the sample mean and the sample variance of independently identically distributed random variables characterizes the Gaussian law and the property of remaining independent under rotations characterizes the Gaussian random variables. In this paper, we consider the free analogue of such a kind of characterizations replacing independence by freeness. We show that freeness of the certain pair of the linear form and the quadratic form in freely identically distributed noncommutative random variables, which covers the case for the sample mean and the sample variance, characterizes the semicircle law. Moreover we give the alternative proof for Nica's result that the property of remaining free under rotations characterizes a semicircular system. Our proof is more direct and straightforward one. Received: 12 February 1997 / Revised version: 16 June 1998