CLT for linear random fields with martingale increments

In [V. Paulauskas, On Beveridge–Nelson decomposition and limit theorems for linear random fields, J. Multivariate Anal., 101:621–639, 2010], limit theorems for linear random fields generated by independent identically distributed innovations were proved. In this paper, which can be regarded as a continuation of the above-mentioned paper, CLT for sums of linear random field are proved in the case where innovations form martingale differences on the plane (that can be defined in several ways). In both papers, the so-called Beveridge–Nelson decomposition is used.     

Normal approximation for linear stochastic processes and random fields in Hilbert space

The central limit theorem is proved for linear random fields defined on an integer-valued lattice of arbitrary dimension and taking values in Hilbert space. It is shown that the conditions in the central limit theorem are optimal. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 421–428, September, 2000.     

Veraverbeke’s theorem at large: on the maximum of some processes with negative drift and heavy tail innovations

Veraverbeke’s (Stoch Proc Appl 5:27–37, 1977) theorem relates the tail of the distribution of the supremum of a random walk with negative drift to the tail of the distribution of its increments, or equivalently, the probability that a centered random walk with heavy-tail increments hits a moving linear boundary. We study similar problems for more general processes. In particular, we derive an analogue of Veraverbeke’s theorem for fractional integrated ARMA models without prehistoric influence, when the innovations have regularly varying tails. Furthermore, we prove some limit theorems for the trajectory of the process, conditionally on a large maximum. Those results are obtained by using a general scheme of proof which we present in some detail and should be of value in other related problems.     

A functional central limit theorem for diffusions on periodic submanifolds of ℝ N

We prove a functional central limit theorem for diffusions on periodic sub- manifolds of ℝ^N. The proof is an adaptation of a method presented in [BenLioPap] and [Bha] for proving functional central limit theorems for diffusions with periodic drift vectorfields. We then apply the central limit theorem in order to obtain a recurrence and a transience criterion for periodic diffusions. Other fields of applications could be heat-kernel estimates, similar to the ones obtained in [Lot].Mathematics Subject Classification (2000): 35B27, 60F05, 58J65The author wants to express his gratitude toward the National Cheng Kung University in Tainan (Taiwan) for its kind hospitality.     

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     

On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria

In this paper we study the central limit theorem and its weak invariance principle for sums of non-adapted stationary sequences, under different normalizations. Our conditions involve the conditional expectation of the variables with respect to a given σ-algebra, as done in Gordin (Dokl. Akad. Nauk SSSR 188, 739–741, 1969) and Heyde (Z. Wahrsch. verw. Gebiete 30, 315–320, 1974). These conditions are well adapted to a large variety of examples, including linear processes with dependent innovations or regular functions of linear processes.     

Almost sure central limit theorem without logarithmic sums

We investigate possible rates of convergence in the almost sure central limit theorem for sums of independent random variables and martingales. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 110–119.     

Some convergence results on quadratic forms for random fields and application to empirical covariances

Limit theorems are proved for quadratic forms of Gaussian random fields in presence of long memory. We obtain a non central limit theorem under a minimal integrability condition, which allows isotropic and anisotropic models. We apply our limit theorems and those of Ginovian (J. Contemp. Math. Anal. 34(2):1?C15) to obtain the asymptotic behavior of the empirical covariances of Gaussian fields, which is a particular example of quadratic forms. We show that it is possible to obtain a Gaussian limit when the spectral density is not in L ². Therefore the dichotomy observed in dimension d?=?1 between central and non central limit theorems cannot be stated so easily due to possible anisotropic strong dependence in d?>?1.     

Central Limit Theorems for Asymptotically Negatively Associated Random Fields

Abstract The aim of this paper is to investigate the central limit theorems for asymptotically negatively dependent random fields under lower moment conditions or the Lindeberg condition. Results obtained improve a central limit theorem of Roussas [11] for negatively assiated fields and the main results of Su and Chi [18], and also include a central limit of theorem for weakly negatively associated random variables similar to that of Burton et al. [20]. Research supported by National Natural Science Foundation of China (No. 19701011)     

On estimation of the remainder in the central limit theorem

A nonuniform estimate of the remainder in the central limit theorem is obtained for a sequence of independent, identically distributed random variables. This estimate is a generalization of an earlier result of L. V. Osipov and the author. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 142–146.     

Rates of convergence in the CLT for linear random fields

In this paper, we study sums of linear random fields defined on the lattice Z ² with values in a Hilbert space. The rate of convergence of distributions of such sums to the Gaussian law is discussed, and mild sufficient conditions to obtain an approximation of order n ^−p are presented. This can be considered as a complement of a recent result of [A.N. Nazarova, Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space, Fundam. Prikl. Mat., 8:1091–1098, 2002 (in Russian)], where the logarithmic rate of convergence was stated, and as a generalization of the result of [D. Bosq, Erratum and complements to Berry–Esseen inequality for linear processes in Hilbert spaces, Stat. Probab. Lett., 70:171–174, 2004] for linear processes.     

The random integral representation conjecture: a quarter of a century later

In [Z.J. Jurek, Relations between the s-selfdecomposable and selfdecomposable measures, Ann. Probab., 13(2):592–608, 1985] and [Z.J. Jurek, Random integral representation for classes of limit distributions similar to Lévy class L ₀, Probab. Theory Relat. Fields, 78:473–490, 1988] the random integral representation conjecture was stated. It claims that (some) limit laws can be written as the probability distributions of random integrals of the form ò_{( a,b ]}h(t)\textdY_v( r(t) ) \int {_{\left( {a,b} \right]}h(t){\text{d}}{Y_v}\left( {r(t)} \right)} for some deterministic functions h, r, and a Lévy process Y_v(t),  t \geqslant 0 {Y_v}(t),\;t \geqslant 0 . Here we review situations where such a claim holds. Each theorem is followed by a remark that gives references to other related papers, results, and historical comments. Moreover, some open questions are stated.     

Berry-Esseen type estimations for multi-sample U-statistics

We study the rate of convergence in the central limit theorem for nondegenerate multi-sample U-statistics of a series of independent samples of independent random variables under minimal sufficient moment conditions on the canonical functions of the Hoeffding representation. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 69–90.     

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     

A Note on the Martingale Method of Proving the Central Limit Theorem for Stationary Sequences

Under appropriate assumptions, the martingale approximation method allows us to reduce the study of the asymptotic behavior of sums of random variables that form a stationary random sequence to a similar problem for sums of stationary martingale differences. In an early paper on the martingale method, the author have proposed certain sufficient conditions for the central limit theorem to hold. It is shown in the present note that these conditions, at least in one particular case, can be essentially relaxed. In the context of the central limit theorem for Markov chains, a similar observation was done in a recent Holzmann and author's work. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 124–132.     

Mixing Limit Theorems of Maximum of Absolute Sums and their Convergence Rates

Conditioned, in the sense of Rényi (Acta Math. Acad. Sci. Hungar. 9, 215–228 1958), limit theorem in the L_p-norm of the maximum of absolute sums of independent identically distributed random variables is established and its exact rate of convergence is given. The results are equivalent to establishing analogous results for the supremum of random functions of partial sums defined on C[0,1], i.e., the invariance principle. New methodologies are used to prove the results that are profoundly different from those used in Rényi (Acta Math. Acad. Sci. Hungar. 9, 215–228, 1958) and subsequent authors in proving the conditioned central limit theorem for partial sums.     

Universal bounds on the selfaveraging of random diffraction measures

 We consider diffraction at random point scatterers on general discrete point sets in ℝ^ν, restricted to a finite volume. We allow for random amplitudes and random dislocations of the scatterers. We investigate the speed of convergence of the random scattering measures applied to an observable towards its mean, when the finite volume tends to infinity. We give an explicit universal large deviation upper bound that is exponential in the number of scatterers. The rate is given in terms of a universal function that depends on the point set only through the minimal distance between points, and on the observable only through a suitable Sobolev-norm. Our proof uses a cluster expansion and also provides a central limit theorem. Received: 10 October 2001 / Revised version: 26 January 2003 / Published online: 15 April 2003 Work supported by the DFG Mathematics Subject Classification (2000): 78A45, 82B44, 60F10, 82B20 Key words or phrases: Diffraction theory – Random scatterers – Random point sets – Quasicrystals – Large deviations – Cluster expansions     

The rationality of vector valued modular forms associated with the Weil representation

 In a recent paper [Duke Math. J., 97, 219–233], Borcherds asks whether or not the spaces of vector valued modular forms associated to the Weil representation have bases of modular forms whose Fourier expansions have only integer coefficients. We give an affirmative answer to Borcherds' question. This strengthens and simplifies Borcherds' main theorem which is a generalization of a theorem of Gross, Kohnen, and Zagier [Math. Ann., 278, 497–562]. Received: 27 September 2001 / Revised version: 22 July 2002 / Published online: 28 March 2003 Mathematics Subject Classification (1991): 11F30; 11F27     

On Ha＇s Version of Set-valued Ekeland＇s Variational Principle

By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187–206 (2005)] established a new version of Ekeland’s variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha’s version of set-valued Ekeland’s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha’s version, we deduce a Caristi-Kirk’s fixed point theorem and a Takahashi’s nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.