Limit Theorem For Stationary Processes Research Articles

Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard

We show how Rio's method [Probab. Theory Related Fields 104 (1996) 255--282] can be adapted to establish a rate of convergence in ${\frac{1}{\sqrt{n}}}$ in the multidimensional central limit theorem for some stationary processes in the sense of the Kantorovich metric. We give two applications of this general result: in the case of the Knudsen gas and in the case of the Sinai billiard.

Moderate deviations for martingale differences and applications to φ -mixing sequences

For a R d -valued sequence of martingale differences { m k } k S 1 , we obtain a moderate deviation principle for the sequence of partial sums { Z n ( t ) 1 ~ k =1 [ nt ] m k / b n , t ] [0,1]}, in the space of càdlàg functions equipped with the Skorohod topology, under the following conditions: a Chen-Ledoux type condition, an exponential convergence in probability of the associated quadratic variation process of the martingale, and a condition of "Lindeberg" type. For the small jumps of Z n (·), we apply the general result of Puhalskii [Puhalskii, A. (1994). "Large deviations of semimartingales via convergence of the predictable characteristics". Stoch. Stoch. Rep. , 49 , pp. 27-85]. Following the method of Ledoux [Ledoux, M. (1992). "Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi". Ann. Inst. H. Poincaré , 28 , pp. 267-280] and Arcones [Arcones, A. (1999). "The large deviation principle for stochastic processes", Submitted for publication], we prove that the large jumps part of Z n (·) is negligible in the sense of the moderate deviations. One can regard our result as an extension to martingale differences, of the beautiful characterization of moderate deviations for i.i.d.r.v. case due to Chen [Chen, X. (1991). "The moderate deviations of independent vectors in Banach space". Chin. J. Appl. Probab. Stat. , 7 , pp. 124-32] and Ledoux [Ledoux, M. (1992). "Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi". Ann. Inst. H. Poincaré , 28 , pp. 267-280]. Using the Gordin [Gordin, M.I. (1969). "The central limit theorem for stationary processes". Soviet Math. Dokl. , 10 , pp. 1174-1176] decomposition, the martingale result is applied to prove the moderate deviation principle for a wide class of stationary { -mixing sequences of random variables.

Statistical analysis for stationary time processes with irregular observations

In a recent paper statistical analysis for stationary time processes with irregular observations was considered. In contrast to known results it is not assumed that all moments exist. Several results on the asymptotic norm behavior of expectation mathematics, covariance and dispersion are derived. Some recent results on the central limit theorem for stationary processes with irregular observations are obtained.

On the functional central limit theorem for stationary processes

We give a sufficient condition for a stationary sequence of square-integrable and real-valued random variables to satisfy a Donsker-type invariance principle. This condition is similar to the L 1 -criterion of Gordin for the usual central limit theorem and provides invariance principles for α-mixing or β-mixing sequences as well as stationary Markov chains. In the latter case, we present an example of a non irreducible and non α-mixing chain to which our result applies.

Correction: A Central Limit Theorem for Stationary Processes and the Parameter Estimation of Linear Processes

Correction: A Central Limit Theorem for Stationary Processes and the Parameter Estimation of Linear Processes

A Functional Limit Theorem for Stationary Processes: A Martingale Approach

Previous article Next article A Functional Limit Theorem for Stationary Processes: A Martingale ApproachD. O. ChikinD. O. Chikinhttps://doi.org/10.1137/1134083PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons Inc., New York, 1968xii+253 38:1718 0172.21201 Google Scholar[2] Walter Philipp and , William Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. 2, (1975), iv+140 55:6570 0361.60007 CrossrefGoogle Scholar[3] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– 48:1287 0219.60027 Google Scholar[4] P. Crepel, Variables aléatoires dépendantsOutils et modèles mathématiques pour 1'automatique: L'analyse de systèmes et le traitement du signal, Vol. 1, Hermann, Paris, 1981, 565–592 0479.60008 Google Scholar[5] P. 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Nauk SSSR, 239 (1978), 766–767, (In Russian.) Google Scholar[11] R. N. Bhattacharya, On the functional central limit theorem and the law of the iterated logarithm for Markov processes, Z. Wahrsch. Verw. Gebiete, 60 (1982), 185–201 83h:60072 0468.60034 CrossrefGoogle Scholar[12] A. Touati, Théorèmes de limite centrale fonctionnels pour les processus de Markov, Ann. Inst. H. Poincaré Sect. B (N.S.), 19 (1983), 43–55 85h:60052 0511.60029 Google Scholar[13] J. de Sam Lazaro, Questions de théorie des flots, Lect. Notes Math., Vol. 465, 1975, 1–96 Google Scholar[14] Yu. A. Rozanov, Stationary random processes, Translated from the Russian by A. Feinstein, Holden-Day Inc., San Francisco, Calif., 1967vi+211 35:4985 0152.16302 Google Scholar[15] R. Sh. Liptser and , A. N. Shiryayev, Theory of martingales, Mathematics and its Applications (Soviet Series), Vol. 49, Kluwer Academic Publishers Group, Dordrecht, 1989xiv+792 90j:60046 0728.60048 CrossrefGoogle Scholar[16] Torgny Lindvall, Weak convergence of probability measures and random functions in the function space $D(0,\,\infty )$, J. Appl. Probability, 10 (1973), 109–121 50:14870 0258.60008 CrossrefGoogle Scholar[17] Philip Protter, Semimartingales and measure preserving flows, Ann. Inst. H. Poincaré Probab. Statist., 22 (1986), 127–147 88m:60138 0598.60046 Google Scholar[18] P. A. Meyer, Probability and potentials, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966xiii+266 34:5119 0138.10401 Google Scholar[19] D. L. McLeish, Functional and random central limit theorems for the Robbins-Monro process, J. Appl. Probability, 13 (1976), 148–154 53:6929 0356.62066 CrossrefGoogle Scholar[20] A. N. Shiryayev, Probability, Graduate Texts in Mathematics, Vol. 95, Springer-Verlag, New York, 1984xi+577, Berlin 85a:60007 0536.60001 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails On the Rate of Convergence of Solution of Differential Equation Disturbed by “Physical” White Noise to a Solution of Appropriate Diffusion Equation. Exponential MixingB. V. Bondarev and S. M. Kozyr15 November 2012 | Theory of Probability & Its Applications, Vol. 56, No. 4AbstractPDF (224 KB)Martingale approximations for continuous-time and discrete-time stationary Markov processesStochastic Processes and their Applications, Vol. 115, No. 9 Cross Ref The central limit theorem for stationary Markov processes with normal generator—with applications to hypergroups20 August 2006 | Stochastics, Vol. 77, No. 4 Cross Ref On the Kolmogorov--Hajek--Rényi Inequality for Normed Integrals of Weakly Dependent ProcessesB. V. Bondarev17 February 2012 | Theory of Probability & Its Applications, Vol. 42, No. 2AbstractPDF (271 KB)Robust Algorithms of the Type of Stochastic Approximation (Continuous Time)S. V. Lototskii12 July 2006 | Theory of Probability & Its Applications, Vol. 40, No. 2AbstractPDF (1674 KB) Volume 34, Issue 4| 1990Theory of Probability & Its Applications History Submitted:29 December 1986Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1134083Article page range:pp. 668-678ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Refinement of the Functional Central Limit Theorem for Stationary Processes

Refinement of the Functional Central Limit Theorem for Stationary Processes

Asymptotic normality of spectral estimates

The asymptotic normality of some spectral estimates, including a functional central limit theorem for an estimate of the spectral distribution function, is proved for fourth-order stationary processes. In contrast to known results it is not assumed that all moments exist or that the process is linear. The data are allowed to be tapered. Using some recent results on the central limit theorem for stationary processes, corollaries are obtained for strong and φ-mixing sequences and linear transformations of martingale differences.

On a Condition of Almost Markov Regularity

Previous article Next article On a Condition of Almost Markov RegularityV. A. StatulyavichusV. A. Statulyavichushttps://doi.org/10.1137/1128030PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] B. Ryauba, The central limit theorem for sums of series of weakly dependent random variables, Litovsk. Mat. Sb., 2 (1962), 193–205, (In Russian.) 29:6526 Google Scholar[2] I. G. Zhurbenko, On the estimation of mixed cumulants for a class of random processes, Theory Prob. Appl., 15 (1970), 525–528 LinkGoogle Scholar[3] I. G. Zhurbenko, On strong estimates of mixed cumulants of random processes, Siberian Math. J., 13 (1972), 302–303 CrossrefGoogle Scholar[4] I. G. Zhurbenko and , N. M. Zuev, The higher spectral densities of stationary processes with mixing, Ukrain. Mat. Ž., 27 (1975), 452–463, 572 53:14620 0325.60039 Google Scholar[5] I. G. Zhurbenko, Mixing conditions and asymptotic theory of stationary time seriesStochastic processes and related topics (Proc. Summer Res. Inst. on Statist. Inference for Stochastic Processes, Indiana Univ., Bloomington, Ind., 1974, Vol. 1; dedicated to Jerzy Neyman), Academic Press, New York, 1975, 259–265 51:9177 0349.60031 Google Scholar[6] V. A. Statulyavichus, Limit theorems for random functions. I, Litovsk. Mat. Sb., 10 (1970), 583–592, (In Russian.) 43:2765 Google Scholar[7] V. A. Statulyavichus, Limit theorems for dependent random variables under various regularity conditions, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, 173–181 55:1420 Google Scholar[8] V. A. Statulyavichus, Application of semi-invariants to asymptotic analysis of distributions of random processesMultivariate analysis, IV (Proc. Fourth Internat. Sympos., Dayton, Ohio, 1975), North-Holland, Amsterdam, 1977, 325–337 57:1614 Google Scholar[9] R. L. Dobrushin, The description of a random field by means of conditional probabilities and conditions for its regularity, Theory Prob. Appl., 13 (1968), 197–224 LinkGoogle Scholar[10] I. A. Ibragimov, Some limit theorems for stationary processes, Theory Prob. Appl., 7 (1962), 349–382 LinkGoogle Scholar[11] M. Rosenblatt, Some remarks on a mixing condition, Ann. Probab., 7 (1979), 170–172 80k:60071a 0925.60052 CrossrefGoogle Scholar[12] V. A. Statulyavichus, Large-deviation theorems for sums of dependent random variables, Lithuanian Math. J., 19 (1979), 289–295 CrossrefGoogle Scholar[13] A. V. Bulinskii, Limit Theorems for Random Processes and Fields, MGU, Moscow, 1981, (In Russian.) Google Scholar[14] V. Statulyavichus, On limit theorems for dependent random variables, Abstracts of Communications of the Third International Vilnius Conference on Probability Theory and Mathematical Statistics, Vilnius: In-t matem. i kibern. AN Lit SSR, 1981, 320–323 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails An Almost-Markov-Type Mixing Condition and Large Deviations for Boolean Models on the LineActa Applicandae Mathematicae, Vol. 96, No. 1-3 | 23 March 2007 Cross Ref Approximation of Distributions of Sums of Weakly Dependent Random Variables by the Normal DistributionLimit Theorems of Probability Theory | 1 Jan 2000 Cross Ref Limit Theorems on Large DeviationsLimit Theorems of Probability Theory | 1 Jan 2000 Cross Ref Every “lower psi-mixing” Markov chain is “interlaced rho-mixing”Stochastic Processes and their Applications, Vol. 72, No. 2 | 1 Dec 1997 Cross Ref Random fields: Applications in cell biologyStochastic Spatial Processes | 16 September 2006 Cross Ref Volume 28, Issue 2| 1984Theory of Probability & Its Applications219-469 History Submitted:10 January 1983Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1128030Article page range:pp. 379-383ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Central Limit Theorem for Stationary Processes in Hilbert Space

Previous article Next article Central Limit Theorem for Stationary Processes in Hilbert SpaceV. V. Mal’tsev and E. I. OstrovskiiV. V. Mal’tsev and E. I. Ostrovskiihttps://doi.org/10.1137/1127036PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout Previous article Next article FiguresRelatedReferencesCited ByDetails Bootstrap for dependent Hilbert space-valued random variables with application to von Mises statisticsJournal of Multivariate Analysis, Vol. 133 | 1 Jan 2015 Cross Ref Weak convergence for the covariance operators of a Hilbertian linear processStochastic Processes and their Applications, Vol. 99, No. 1 | 1 May 2002 Cross Ref Multilinear forms and measures of dependence between random variablesJournal of Multivariate Analysis, Vol. 16, No. 3 | 1 Jun 1985 Cross Ref A note on the convergence to Gaussian laws of sums of stationary φ-mixing triangular arraysProbability in Banach Spaces V | 16 September 2006 Cross Ref Volume 27, Issue 2| 1983Theory of Probability & Its Applications215-440 History Submitted:04 July 1979Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1127036Article page range:pp. 357-359ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

A Central Limit Theorem for Stationary Processes and the Parameter Estimation of Linear Processes

A central limit theorem is proved for the sample covariances of a linear process. The sufficient conditions for the theorem are described by more natural ones than usual. We apply this theorem to the parameter estimation of a fitted spectral model, which does not necessarily include the true spectral density of the linear process. We also deal with estimation problems for an autoregressive signal plus white noise. A general result is given for efficiency of Newton-Raphson iterations of the likelihood equation.

Gordin's theorem and the periodogram

In 1968 M. I. Gordin proved a very strong central limit theorem for stationary, ergodic sequences by means of approximation with martingales. In the present paper Gordin's theorem is generalized to cover also the periodogram of a stationary sequence, and the restriction of ergodicity is removed. It is noted that known central limit theorems for stationary processes can often be generalized to the periodogram by means of this result.

Gordin's theorem and the periodogram

In 1968 M. I. Gordin proved a very strong central limit theorem for stationary, ergodic sequences by means of approximation with martingales. In the present paper Gordin's theorem is generalized to cover also the periodogram of a stationary sequence, and the restriction of ergodicity is removed. It is noted that known central limit theorems for stationary processes can often be generalized to the periodogram by means of this result.

On Gordin's central limit theorem for stationary processes

The central limit theorem for ergodic stationary processes obtained by Gordin is shown to hold for general stationary processes. In this case, the limit law is a mixture of normals.

On Gordin's central limit theorem for stationary processes

The central limit theorem for ergodic stationary processes obtained by Gordin is shown to hold for general stationary processes. In this case, the limit law is a mixture of normals.

Central Limit Theorem for a Class of Billiards

Previous article Next article Central Limit Theorem for a Class of BilliardsL. A. BunimovicL. A. Bunimovichttps://doi.org/10.1137/1119006PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] L. A. Bunimovič, The ergodic properties of billiards that are nearly scattering, Dokl. Akad. Nauk SSSR, 211 (1973), 1024–1026, (In Russian.) MR0330413 (48:8750) Google Scholar[2] A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR (N.S.), 119 (1958), 861–864, (In Russian.) MR0103254 (21:2035a) 0083.10602 Google Scholar[3] V. A. Rokhlin, Recent progress in the theory of transformations with invariant measure, Uspekhi Mat. Nauk., 15 (1960), 3–26, (In Russian.) Google Scholar[4] Ja. G. Sinai˘, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Uspehi Mat. Nauk, 25 (1970), 141–192, (In Russian.) MR0274721 (43:481) Google Scholar[5] L. A. Bunimovich, A central limit theorem for scattering billiards, Dokl. Akad. Nauk SSSR, 204 (1972), 778–781 MR0301167 (46:325) Google Scholar[6] V. A. Rokhlin, On the basic concepts of measure theory, Mat. Sb., 25 (67) (1949), 107–150, (In Russian.) Google Scholar[7] L. A. Bunimovich and , Ya. G. Sinai, The fundamental theorem of the theory of scattering billiards, Mat. Sb. (N.S.), 90(132) (1973), 415–431, 479, (In Russian.) MR0367153 (51:3395) Google Scholar[8] Ya. G. Sinai, Classical dynamic systems with countably-multiple Lebesgue spectrum. II, Izv. Akad. Nauk SSSR Ser. Mat., 30 (1966), 15–68 MR0197684 (33:5847) Google Scholar[9] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209–, “Nauka”, Moscow, (In Russian.) MR0224110 (36:7157) Google Scholar[10] D. V. Anosov and , ya. G. Sinai, Certain smooth ergodic systems, Uspehi Mat. Nauk, 22 (1967), 107–172, (In Russian.) MR0224771 (37:370) 0177.42002 Google Scholar[11] Ya. G. Sinai, Central limit theorem for geodesic flows on manifolds of constant negative curvature, Dokl. Akad. Nauk SSSR, 133 (1960), 1303–1306, (In Russian.) Google Scholar[12] Ya. G. Sinai, Candidate's Dissertation, Moscow State Univ., 1960 Google Scholar[13] Ya. G. Sinai, On limit theorems for stationary processes, Theory Prob. Applications, 7 (1962), 205–210 10.1137/1107021 0115.35804 LinkGoogle Scholar[14] M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Nat. Acad. Sci. U. S. A., 42 (1956), 43–47 MR0074711 (17,635b) 0070.13804 CrossrefGoogle Scholar[15] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– MR0322926 (48:1287) 0219.60027 Google Scholar[16] B. V. Gnedenko and , A. N. Kolmogorov, Limit distributions for sums of independent random variables, Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1954ix+264 MR0062975 (16,52d) 0056.36001 Google Scholar[17] V. P. Leonov, Applications of Higher Order Cumulants to the Theory of Stationary Stochastic Processes, Nauka, Moscow, 1964, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Dynamical systems with time scale separation: averaging, stochastic modelling, and central limit theorems Cross Ref Approach to Gaussian stochastic behavior for systems driven by deterministic chaotic forces1 November 1999 | Physical Review E, Vol. 60, No. 5 Cross Ref Dynamical systems of Langevin typePhysica A: Statistical Mechanics and its Applications, Vol. 233, No. 1-2 Cross Ref Brownian motion from deterministic dynamicsPhysica A: Statistical Mechanics and its Applications, Vol. 169, No. 2 Cross Ref Transport properties of stochastic Lorentz models1 January 1982 | Reviews of Modern Physics, Vol. 54, No. 1 Cross Ref Long-time correlation effects on displacement distributionsJournal of Statistical Physics, Vol. 19, No. 4 Cross Ref Lecture VIII a model for couette flow data27 August 2006 Cross Ref Computer results on transport properties Cross Ref Volume 19, Issue 1| 1974Theory of Probability & Its Applications History Submitted:26 January 1973Published online:28 July 2006 InformationCopyright © 1974 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1119006Article page range:pp. 65-85ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On the central limit theorem for stationary processes

On the central limit theorem for stationary processes

On Stochastic Approximation for Random Processes with Continuous Time

Previous article Next article On Stochastic Approximation for Random Processes with Continuous TimeT. P. KrasulinaT. P. Krasulinahttps://doi.org/10.1137/1116073PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Herbert Robbins and , Sutton Monro, A stochastic approximation method, Ann. Math. Statistics, 22 (1951), 400–407 MR0042668 0054.05901 CrossrefGoogle Scholar[2] J. Kiefer and , J. Wolfowitz, Stochastic estimation of the maximum of a regression function, Ann. Math. Statistics, 23 (1952), 462–466 MR0050243 0049.36601 CrossrefGoogle Scholar[3] Miloslav Driml and , J. Nedoma, Stochastic approximations for continuous random processes, Trans. 2nd Prague Conf. Information Theory, Publ. House Czechoslovak Acad. Sci., Prague, Academic Press, New York, 1960, 145–158 MR0126877 0098.32202 Google Scholar[4] Jerome Sacks, Asymptotic distribution of stochastic approximation procedures, Ann. Math. Statist., 29 (1958), 373–405 MR0098427 0229.62010 CrossrefGoogle Scholar[5] David J. Sakrison, A continuous Kiefer-Wolfowitz procedure for random processes, Ann. Math. Statist., 35 (1964), 590–599 MR0161445 0131.35905 CrossrefGoogle Scholar[6] V. A. Volkonskii˘ and , Yu. A. Rozanov, Some limit theorems for random functions. I, Theor. Probability Appl., 4 (1959), 178–197 MR0121856 0092.33502 LinkGoogle Scholar[7] I. A. Ibragimov, Some limit theorems for stationary processes, Theory Prob. Applications, 7 (1962), 349–382 10.1137/1107036 0119.14204 LinkGoogle Scholar[8] S. G. Mikhlin, Lectures on Linear Integral Equations, Fizmatgiz, Moscow, 1959, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Gradient procedures for stochastic approximation with dependent noise and their asymptotic behaviourInternational Journal of Systems Science, Vol. 16, No. 8 | 10 May 2007 Cross Ref Some Limit Theorems for Processes with Random TimeA. N. BorodinTheory of Probability & Its Applications, Vol. 24, No. 4 | 17 July 2006AbstractPDF (1390 KB)Sufficient conditions for convergence of stochastic approximation algorithms for random processes with continuous timeCybernetics, Vol. 15, No. 6 | 1 Nov 1979 Cross Ref A Stochastic Approximation Procedure in the Case of Weakly Dependent ObservationsA. N. BorodinTheory of Probability & Its Applications, Vol. 24, No. 1 | 17 July 2006AbstractPDF (1321 KB)Recursive stochastic gradient procedures in the presence of dependent noiseStochastic Optimization Cross Ref Volume 16, Issue 4| 1971Theory of Probability & Its Applications575-746 History Submitted:07 July 1969Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1116073Article page range:pp. 674-682ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

Asymptotic Normality of the Number of Crossings of Level Zero by a Gaussian Process

Asymptotic Normality of the Number of Crossings of Level Zero by a Gaussian Process