Mixing normal approximations of vectors of sums and maximum sums

Summary The conditioned central limit theorem for the vector of maximum partial sums based on independent identically distributed random vectors is investigated and the rate of convergence is discussed. The conditioning is that of Rényi (1958,Acta Math. Acad. Sci. Hungar.,9, 215–228). Analogous results for the vector of partial sums are obtained. University of Petroleum and Minerals     

On some limit theorems for the maximum of sums of independent random processes

We investigate conditions for the weak convergence of the maximum of sums of independent random processes in the space L _p and present several applications to the asymptotic analysis of certain ω ²-type statistics. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1664–1674, December, 2008.     

Conditioned rates of convergence in the CLT for sums and maximum sums

An interesting recent result of Landers and Roggé (1977, Ann. Probability5, 595–600) is investigated further. Rates of convergence in the conditioned central limit theorem are developed for partial sums and maximum partial sums, with positive mean and zero mean separately, of sequences of independent identically distributed random variables. As corollaries we obtain a conditioned central limit theorem for maximum partial sums both for positive and zero mean cases.     

Rates of approximation in the multidimensional invariance principle for sums of i.i.d. random vectors with finite moments

The aim of this paper is to derive consequences of a result of G?tze and Zaitsev (2008). We show that the i.i.d. case of this result implies a multidimensional version of some results of Sakhanenko (1985). We establish bounds for the rate of strong Gaussian approximation of sums of i.i.d. R ^d -valued random vectors ξ _j having finite moments E IIξ _j II^γ, γ>2. Bibliography: 13 titles.     

Testing Epidemic Changes of Infinite Dimensional Parameters

To detect epidemic change in the mean of a sample of size n of random elements in a Banach space, we introduce new test statistics DI based on weighted increments of partial sums. We obtain their limit distributions under the null hypothesis of no change in the mean. Under alternative hypothesis our statistics can detect very short epidemics of length log^γ n, γ > 1. We present applications to detect epidemic changes in distribution function or characteristic function of real valued observations as well as changes in covariance matrices of random vectors. Final version 27 October 2004     

The influence of sample extremes on the maximum of partial sums of independent random variables

Many papers exist dealing with the distribution of the maximum of partial sums of independent random variables and studying the relationship between the sums and the sample extremes (see [L. Takacs,Combinator Methods in the Theory of Stochastic Processes, Wiley, New York (1967)], [J. Calambos,The Asymptotic Theory of Extreme Order Statistics, Wiley, New York (1978)]). The present paper deals with the effect the sample extremes have on the distribution of the maximum of sums of independent random variables. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993     

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.     

Convolution with oscillating kernels: Bochner-Riesz and spherical partial sums operators

The paper is an up to date survey of results on theL ^p boundedness, 1 ≤p ≤ ∞, of convolution operators with oscillating kernels, mainly Bochner-Riesz and the spherical partial sums operators in ℝ ⁿ ,n ≥ 2 and their maximal operators. Consequences are norm convergence and almost everywhere convergence of Fourier series. Il lavoro è una rassegna di risultati, fino ai più recenti, sulla limitatezzaL ^p , 1 ≤p ≤ ∞, di operatori di convoluzione con nuclei oscillanti, con particolare riguardo agli operatori di Bochner-Riesz e agli operatori somme parziali sferiche. Come applicazione si hanno teoremi di convergenza in norma e quasi ovunque per serie di Fourier.

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(Conferenza tenuta il 27 giugno 1988)     

A note on rates of convergence in the multidimensional CLT for maximum partial sums

In this note we obtain rates of convergence in the central limit theorem for certain maximum of coordinate partial sums of independent identically distributed random vectors having positive mean vector and a nonsingular correlation matrix. The results obtained are in terms of rates of convergence in the multidimensional central limit theorem. Thus under the conditions of Sazonov (1968, Sankhya, Series A30 181–204, Theorem 2), we have the same rate of convergence for the vector of coordinate maximums. Other conditions for the multidimensional CLT are also discussed, c.f., Bhattachaya (1977, Ann. Probability 5 1–27). As an application of one of the results we obtain a multivariate extension of a theorem of Rogozin (1966, Theor. Probability Appl. 11 438–441).     

Deconvolving Multivariate Density from Random Field

We consider the problem of estimating a continuous bounded multivariate probability density function (pdf) when the random field X _i , i ∈ Z ^d from the density is contaminated by measurement errors. In particular, the observations Y _i , i ∈ Z ^d are such that Y _i = X _i + ε _i , where the errors ε _i are a sample from a known distribution. We improve the existing results in at least two directions. First, we consider random vectors in contrast to most existing results which are only concerned with univariate random variables. Secondly, and most importantly, while all the existing results focus on the temporal cases (d = 1), we develop the results for random vectors with a certain spatial interaction. Precise asymptotic expressions and bounds on the mean-squared error are established, along with rates of both weak and strong consistencies, for random fields satisfying a variety of mixing conditions. The dependence of the convergence rates on the density of the noise field is also studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Inequalities of Maximum of Partial Sums and Weak Convergence for a Class of Weak Dependent Random Variables

In this paper, we establish a Rosenthal-type inequality of the maximum of partial sums for ρ^- -mixing random fields. As its applications we get the Hájeck -Rènyi inequality and weak convergence of sums of ρ^- -mixing sequence. These results extend related results for NA sequence and p^＊ -mixing random fields,     

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.     

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.     

Convergence in law of random sums with non-random centering

We extend results obtained in Kruglov,⁽⁷⁾ and Finkelstein and Tucker⁽³⁾ to obtain necessary and sufficient conditions for convergence in law of random sums of non-identically distributed independent random variables under non-random centering. Thei.i.d. case is also considered for random variables attracted to a stable law. Necessary and sufficient conditions for convergence in law of these random variables under non-random centering, and in some cases, under non-random norming, are also obtained. The distribution functions for the limit laws are determined as well, generalizing results of Robbins.⁽¹⁰⁾ Supported in part by The State University of New York and United States Information Agency Grant No. IA AEMP69193692.     

Estimates for the rate of strong Gaussian approximation for sums of i.i.d. multidimensional random vectors

The aim of this paper is to derive new optimal bounds for the rate of strong Gaussian approximation of sums of i.i.d. R ^d-valued random variables ξ_j that have finite moments of the form EH (‖ξ_j‖), where H (x) is a monotone function growing not slower than x² and not faster than e^cx. We obtain some generalization and improvements of results of U. Einmahl (1989). Bibliography: 28 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 141–158.     

A shorter proof of Kanter’s Bessel function concentration bound

We give a shorter proof of Kanter’s (J. Multivariate Anal. 6, 222–236, 1976) sharp Bessel function bound for concentrations of sums of independent symmetric random vectors. We provide sharp upper bounds for the sum of modified Bessel functions I₀(x) + I₁(x), which might be of independent interest. Corollaries improve concentration or smoothness bounds for sums of independent random variables due to Čekanavičius & Roos (Lith. Math. J. 46, 54–91, 2006); Roos (Bernoulli, 11, 533–557, 2005), Barbour & Xia (ESAIM Probab. Stat. 3, 131–150, 1999), and Le Cam (Asymptotic Methods in Statistical Decision Theory. Springer, Berlin Heidelberg New York, 1986).      

Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications

In this paper, we establish some Rosenthal type inequalities for maximum partial sums of asymptotically almost negatively associated random variables, which extend the corresponding results for negatively associated random variables. As applications of these inequalities, by employing the notions of residual Cesàro α-integrability and strong residual Cesàro α-integrability, we derive some results on L _p convergence where 1 < p < 2 and complete convergence. In addition, we estimate the rate of convergence in Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.     

Asymptotics of multinomial sums and identities between multi-integrals

We calculate the asymptotics of combinatorial sums ∑ _α f(α)( _α ⁿ ) ^β , whereα = (α ₁, …,α _h ) withα _i =α _j for certaini, j. Hereh is fixed and theα _i ’s are natural numbers. This implies the asymptotics of the correspondingS _n -character degrees ∑_λ f(λ)d _λ ^β . For certain sequences ofS _n characters which involve Young’s rule, the latter asymptotics were obtained earlier [1] by a different method. Equating the two asymptotics, we obtain equations between multi-integrals which involve Gaussian measures. Special cases here give certain extensions of the Mehta integral [5], [6]. Supported by the Weizmann Institute of Science, Rehovot, Israel; by the Institute for Advanced Study, Princeton, New Jersey, USA; NSF grant number DMS 9304580; and by the Centre National de Recherche Scientifique, Lille, France. This work was partially supported by an NSF grant number DMS 94-01197.     

The convergence of partial sums of interpolating polynomials

For functions of ΛBV, we study the convergence of the partial sums of interpolating polynomials. An estimate is found for the Fourier-Lagrange coefficients of these functions. For functions in BV, convergence is shown at points of discontinuity if the order of the polynomial increases sufficiently rapidly compared to the order of the partial sum. A Dirichlet-Jordan type theorem is shown for functions of harmonic bounded variation, and this result is shown to be best possible.     

On the strong convergence of weighted sums

Let {X _n ,n≥1} be a sequence of independent and identically distributed random variables and {a _ni ,1≤i≤n,n≥1} an array of constants. Some strong convergence results for the weighted sums ∑ _i=1 ⁿ a _ni X _i are obtained.