Key points are not available for this paper at this time.
Previous article Next article Self-Similar Probability DistributionsYu. G. SinaiYu. G. Sinaihttps: //doi. org/10. 1137/1121005PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout1 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 0056. 36001 Google Scholar2 R. L. Dobrušin, Gibbsian random fields for lattice systems with pairwise interactions. , Funkcional. Anal. i Priložen. , 2 (1968), 31–43 MR0250630 Google Scholar3 R. L. Dobrušin, The problem of uniqueness of a Gibbsian random field and the problem of phase transitions, Funkcional. Anal. i Priložen. , 2 (1968), 44–57, (In Russian. ) MR0250631 Google Scholar4 K. Itô, Stochastic Processes, Aarhus University, 1969 0226. 60053 Google Scholar5 A. N. Kolmogorov, Wiener's helix and other interesting curves in Hilbert space, Dokl. Akad. Nauk SSSR, 26 (1940), 115–118, (In Russian. ) Google Scholar6 John Lamperti, Probability. A survey of the mathematical theory, W. A. Benjamin, Inc. , New York-Amsterdam, 1966x+150 MR0206996 0147. 15502 Google Scholar7 M. Fisher, General scaling theory for critical point, Proc. of the Nobel Symp. , Vol. 24, 1973, 16–37 Google Scholar8 Kiyosi Itô, Multiple Wiener integral, J. Math. Soc. Japan, 3 (1951), 157–169, Japan MR0044064 0044. 12202 CrossrefGoogle Scholar9 Kiyosi Itô, Complex multiple Wiener integral, Jap. J. Math. , 22 (1952), 63–86 (1953) MR0063609 0049. 08602 CrossrefGoogle Scholar10 O. E. Lanford, III and, D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics, Comm. Math. Phys. , 13 (1969), 194–215 10. 1007/BF01645487 MR0256687 CrossrefGoogle Scholar11 K. Wilson and, J. Kogut, The renormalization group and the ε-expansion, preprint, Inst. for Adv. Studies, Princeton, New Jersey, 1972 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Entropic Compressibility of Lévy ProcessesIEEE Transactions on Information Theory, Vol. 68, No. 8 Cross Ref Functional Limit Theorems for the Fractional Ornstein–Uhlenbeck Process22 October 2020 | Journal of Theoretical Probability, Vol. 35, No. 1 Cross Ref Zeros of Gaussian power series, Hardy spaces and determinantal point processes13 December 2021 | Annals of Functional Analysis, Vol. 13, No. 1 Cross Ref On the inverse problem of fractional Brownian motion and the inverse of infinite Toeplitz matrices13 October 2021 | Journal of Physics Communications Cross Ref A Generalization of Pisier Homogeneous Banach AlgebraMichigan Mathematical Journal, Vol. -1, No. -1 Cross Ref Rough Homogenisation with Fractional Dynamics1 January 2022 Cross Ref Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises9 February 2018 | Journal of Theoretical Probability, Vol. 32, No. 3 Cross Ref Asymptotically efficient estimators for self-similar stationary Gaussian noises under high frequency observationsBernoulli, Vol. 25, No. 3 Cross Ref A Self‐Normalized Semi‐Parametric Test to Detect Changes in the Long Memory Parameter4 January 2019 | Journal of Time Series Analysis, Vol. 40, No. 4 Cross Ref Constant-Length Random Substitutions and Gibbs Measures10 March 2018 | Journal of Statistical Physics, Vol. 171, No. 2 Cross Ref Theoretical and Numerical Comparisons of the Parameter Estimator of the Fractional Brownian Motion1 November 2018 Cross Ref On the validity of resampling methods under long memoryThe Annals of Statistics, Vol. 45, No. 6 Cross Ref Processes on the emergent landscapes of biochemical reaction networks and heterogeneous cell population dynamics: differentiation in living matters10 May 2017 | Journal of The Royal Society Interface, Vol. 14, No. 130 Cross Ref The locality phenomenon in the analysis of self-similar network traffic flows Cross Ref Fractals1 December 2015 Cross Ref Asymptotic equivalence for regression under fractional noiseThe Annals of Statistics, Vol. 42, No. 6 Cross Ref Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt processJournal of Multivariate Analysis, Vol. 131 Cross Ref A Fast Multifractal Model for Self-Similar Traffic Flows in High-Speed Computer NetworksProcedia Computer Science, Vol. 17 Cross Ref A new process for modeling heartbeat signals during exhaustive run with an adaptive estimator of its fractal parametersJournal of Applied Statistics, Vol. 39, No. 6 Cross Ref The Effect of Round-off Error on Long Memory ProcessesSSRN Electronic Journal Cross Ref Modified periodogram method for estimating the Hurst exponent of fractional Gaussian noise17 December 2009 | Physical Review E, Vol. 80, No. 6 Cross Ref Analysis of direct spectrum measurement of a sinusoidal signal impaired by either fractional Gaussian phase noise or fractional Brownian phase motionIEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Vol. 56, No. 11 Cross Ref LÉvy Flights and Fractal Modeling of Internet TrafficIEEE/ACM Transactions on Networking, Vol. 17, No. 1 Cross Ref Does the Internet Still Demonstrate Fractal Nature? Cross Ref Long-range dependencies in algorithmic computing Cross Ref Long-Range Dependence Models for the Analysis and Discrimination of Sea-Surface Anomalies in Sea SAR Imagery19 November 2009 Cross Ref Long memory options: LM evidence and simulationsResearch in International Business and Finance, Vol. 21, No. 2 Cross Ref On Rate-Distortion Models for Natural Images and Wavelet Coding PerformanceIEEE Transactions on Image Processing, Vol. 16, No. 5 Cross Ref An empirical comparison of generators for self similar simulated trafficPerformance Evaluation, Vol. 64, No. 2 Cross Ref Estimating the long memory granger causality effect with a spectrum estimator1 January 2006 | Journal of Forecasting, Vol. 25, No. 3 Cross Ref A bootstrap causality test for covariance stationary processesJournal of Econometrics, Vol. 126, No. 1 Cross Ref Fractional Gaussian noise, functional MRI and Alzheimer's diseaseNeuroImage, Vol. 25, No. 1 Cross Ref Long Memory Options: ValuationSSRN Electronic Journal Cross Ref Long Memory Options: LM Evidence and SimulationsSSRN Electronic Journal Cross Ref References21 January 2010 Cross Ref The nature of discrete second-order self-similarity22 February 2016 | Advances in Applied Probability, Vol. 35, No. 02 Cross Ref Stochastic models and statistical analysis for clock noise6 June 2003 | Metrologia, Vol. 40, No. 3 Cross Ref The nature of discrete second-order self-similarity22 February 2016 | Advances in Applied Probability, Vol. 35, No. 2 Cross Ref The spectra and periodograms of anti-correlated discrete fractional Gaussian noisePhysica A: Statistical Mechanics and its Applications, Vol. 322 Cross Ref Superposition of Diffusions with Linear Generator and its Multifractal Limit Process15 May 2003 | ESAIM: Probability and Statistics, Vol. 7 Cross Ref Fractional Brownian Motion and Fractional Gaussian Noise6 June 2003 Cross Ref Consistent order selection with strongly dependent data and its application to efficient estimationJournal of Econometrics, Vol. 110, No. 2 Cross Ref Gaussian Limit for Determinantal Random Point FieldsThe Annals of Probability, Vol. 30, No. 1 Cross Ref Renormalization group and probability theoryPhysics Reports, Vol. 352, No. 4-6 Cross Ref Nonparametric Test for Causality with Long-range DependenceEconometrica, Vol. 68, No. 6 Cross Ref First passage time distribution for anomalous diffusionPhysics Letters A, Vol. 273, No. 5-6 Cross Ref Communication Network with Self-Similar Traffic Cross Ref Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motionThe Journal of Fourier Analysis and Applications, Vol. 5, No. 5 Cross Ref Stochastic fractal behavior in concentration fluctuation and fluorescence correlation spectroscopyBiophysical Chemistry, Vol. 80, No. 1 Cross Ref Lévy-stability-under-addition and fractal structure of markets: Implications for the investment management industry and emphasized examination of MATIF notional contractMathematical and Computer Modelling, Vol. 29, No. 10-12 Cross Ref Self-similar processes in communications networksIEEE Transactions on Information Theory, Vol. 44, No. 5 Cross Ref On two-dimensional fractional Brownian motion and fractional Brownian random field1 January 1999 | Journal of Physics A: Mathematical and General, Vol. 31, No. 28 Cross Ref Characteristic length scales of spatial models in ecology via fluctuation analysis29 November 1997 | Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, Vol. 352, No. 1361 Cross Ref On self-similar traffic in ATM queues: definitions, overflow probability bound, and cell delay distributionIEEE/ACM Transactions on Networking, Vol. 5, No. 3 Cross Ref Long memory processes and fractional integration in econometricsJournal of Econometrics, Vol. 73, No. 1 Cross Ref Averaged periodogram estimation of long memoryJournal of Econometrics, Vol. 73, No. 1 Cross Ref The Estimation of Continuous Parameter Long-Memory Time Series Models11 February 2009 | Econometric Theory, Vol. 12, No. 2 Cross Ref Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theoryJournal of Statistical Physics, Vol. 72, No. 5-6 Cross Ref Linear Models with Long-Range Dependence and with Finite or Infinite Variance Cross Ref Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regressionJournal of Econometrics, Vol. 47, No. 1 Cross Ref Some Research in Ergodic Theory and Mathematical Problems of Statistical Mechanics in the Moscow University Department of Probability TheoryV. A. Malyshev and Ya. G. Sinai17 July 2006 | Theory of Probability & Its Applications, Vol. 34, No. 1AbstractPDF (1243 KB) A CENTRAL LIMIT THEOREM OF FOURIER TRANSFORMS OF STRONGLY DEPENDENT STATIONARY PROCESSESJournal of Time Series Analysis, Vol. 10, No. 4 Cross Ref Non-Gibbsian limit for large-block majority-spin transformationsJournal of Statistical Physics, Vol. 55, No. 1-2 Cross Ref Renormalisation of a hierarchical ϕ 34 model1 January 1999 | Journal of Physics A: Mathematical and General, Vol. 21, No. 8 Cross Ref Central limit theorems for quadratic forms in random variables having long-range dependenceProbability Theory and Related Fields, Vol. 74, No. 2 Cross Ref A Bibliographical Guide to Self-Similar Processes and Long-Range Dependence Cross Ref Stochastic Measure Processes Cross Ref Scale limit and low momentum behavior of Euclidean fields. IIJournal of Mathematical Physics, Vol. 21, No. 4 Cross Ref Scaling transformations for 0, 1-valued sequencesZeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 53, No. 1 Cross Ref Convergence of integrated processes of arbitrary Hermite rankZeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 50, No. 1 Cross Ref Probabilistic formulation of the problem Cross Ref Statistics of natural images and models Cross Ref Statistics of range images Cross Ref Volume 21, Issue 1| 1976Theory of Probability & Its Applications History Submitted: 24 December 1974Published online: 17 July 2006 InformationCopyright © 1976 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI: 10. 1137/1121005Article page range: pp. 64-80ISSN (print): 0040-585XISSN (online): 1095-7219Publisher: Society for Industrial and Applied Mathematics
Yu. G. Sinai (Wed,) studied this question.