Gaussian Random Functions M.A. Lifshits (Contents of the book)

1. Gaussian Distributions and Random Variables: Gaussian distribution in R1 . Properties of the Gaussian density. Moments. Laplace transform and Fourier transform. Stability. The problem of convexity of Gaussian distribution. Gaussian random variables. The transformation of the properties of Gaussian distributions into the properties of Gaussian random variables. The estimates of Gaussian tail probabilities. 2. Multi-dimensional Gaussian Distributions: Measures in Rn . One-dimensional projections, characteristic functional. Barycentre. Covariance operator. Description of Gaussian distributions in Rn . Gaussian distribution as the linear image of the standard Gaussian distribution. The norm distribution. The approximation of the standard Gaussian distribution by the uniform distribution on the sphere. Gaussian random vector. The joint distribution of the set of random variables. 3. Covariance: Covariance. Independence and absence of correlation. The linear prediction. Geometrical sense of correlation. Independence of non-correlated Gaussian random variables. Linear prediction as the optimal general prediction for Gaussian random vector. 4. Random Functions: Random functions, processes and fields. Modifications and sample functions. Correlation functions. Positive definite functions. The linear space generated by the random function. Separability. Construction of the separable modification. Stationary processes. Processes with stationary, noncorrelated, independent increments. Bochner theorem on the spectral representation of covariance function. Existence of Gaussian random function with given covariance. Covariance conditions providing the stationarity of the process or the stationarity of its increments. 5. Examples of Gaussian Random Functions: Wiener process: history, definition , stationarity and independence of increments, self-similarity. Brownian bridge and its connection with the linear interpolation of Wiener process. Multi-parametric generalizations of the Wiener

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process: Brownian sheet and Levy’s Brownian function. Fractional Brownian motion. White noise as Gaussian random function and a random finitelyadditive measure. Integration over white noise. Spectral representation of a stationary process. Ornstein-Uhlenbeck process. 6. Correlation Models: Notion of the model. Representation of the random function as an integral over white noise. A model for Brownian sheet. Special cases: models for Wiener process and Brownian bridge. A model for Levy’s Brownian motion on L1 , on the metric space embedable in L1 ; on Rn (Chentsov theorem); over the normed space with summing norm. 7. Oscillation: Oscillation of a deterministic function. Properties of oscillations (semicontinuity, subadditivity, concavity). Oscillation of random function. Ito-Nisio theorem on the oscillation determinism. Example: oscillation of the sequence of independent random variables. Belyaev alternative for stationary processes. Transformation of the random function into continuous one by the change of metric. Connection between oscillation and boundedness. 8. Infinite-Dimensional Gaussian Distributions: Distribution of a random function. Locally convex spaces. Duality. Radon measures. Barycentre. Covariance operator. Characteristic functional. Definition of the Gaussian measure. Characteristic functional of the Gaussian measure. Existence of the barycentre and covariance operator. Examples : Gaussian measures in Rn and in Hilbert space. 9. Linear Functionals, Admissible Shifts, and the Kernel: Measurable linear functionals. Admissible shifts and directions. Canonical correspondence between the measurable linear functionals and admissible shifts. Separability of the space of linear functionals. The density of the shifted Gaussian measure. Measure of admissibility of the shift and its calculation by means of the factorization of the covariance operator. Invariant sets and 0-1 laws. Topological and linear support of the Gaussian measure. 10. Some Important Gaussian Distributions: The standard Gaussian measure in R∞ . Gaussian measure in Hilbert space. Wiener measure. The distribution of a stationary process. 11. Convexity and Isoperimetric Property: Gaussian symmetrizations and their properties. Ehrhard’s theorem on the regularity of symmetrizations. Isoperimetric property of halfspaces. Isoperimetric inequalities. Symmetrization and convexity. Logarithmic convexity. The variance ellipsoid. Isoperimetric inequality and Ehrhard inequality for general Gaussian distributions in Rn . Infinite-dimensional versions of these inequalities. The structure of the distribution of a convex functional. 2

12. Large Deviations: The definition of large deviations. General form of logarithmic asymptotic of tail probabilities of Gaussian suprema. Concentration principle. Relations between oscillation and large deviations. Large deviations of Gaussian measure and their relations to covariance operator and action functional. Probabilistic and integral form of Bahadur’s large deviations principle. Estimates for the density of large deviations. Relations between the density and the probability of large deviations. 13. Exact Asymptotic Behavior of Large Deviation Probabilities: Laplace transform and the exact asymptotic of large deviations. Example: exact asymptotic of `p -norm of the Gaussian vector with independent components (1 ≤ p ≤ 2). Large deviations in Hilbert space. Simple form of large deviation asymptotic for locally smooth seminorm with finite set of critical points.

14. Metric Entropy and Comparison Principle: Basic notions: net, metric entropy, Dudley integral. The upper bound for the expectation of supremum by Dudley integral. The entropy estimate of tail probabilities of Gaussian suprema. Examples: the estimate of tail probabilities of the stationary fields with polynomial and logarithmic increments. Fernique integral and corresponding upper estimates of Gaussian random function. Comparison identity. Comparison principle in Schl¨afli-Slepian form. Comparison principle in Sudakov-Fernique form. Metric capacity and lower bounds of r.f. Sudakov minoration. Capacity lower bound for Gaussian extrema. Examples: lower bounds for random fields with polynomial and logarithmic increments. Talagrand minoration.Double sum method and the corresponding lower bounds. Exact asymptotic behavior of Gaussian suprema of regular Gaussian random fields. Examples : cosine-field, several fields related to Wiener process (Brownian sheet, Brownian pillow, set-indexed Brownian sheet). 15. Boundedness and Continuity: Bounded and continuous modifications of Gaussian random functions. Entropy and continuity: Dudley integral as a modulus of continuity of the sample paths. Necessary entropy conditions for boundedness and continuity. Necessary and sufficient conditions for continuity of the homogeneous Gaussian random function (Fernique theorem). 16. Majorizing Measures: Two definitions of majorizing measure. Majorizing measures and metric entropy. Upper and lower bounds of Gaussian extrema expectation by means of majorizing measure. Sufficient and necessary conditions for boundedness and continuity. 17. Functional Law of the Iterated Logarithm:

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Strassen law for Wiener process. Generalizations: the law for arbitrary normalizing functions, the functional law for subsequences. Strassen type laws for strong topologies. R´ev´esz functional law for the increments of Wiener process. Book-Shore effect. Convergence rate for Strassen law. 18. Small Deviations: The definition of small deviation (or small balls) problem. The difference between the results on the large and small deviations.Markov case. Exact and asymptotic formulae for Wiener measure of small balls. The role of differential equations and Laplace transform. The case of stationary Gaussian process Tsyrelson spectral upper estimate for the probabilities of small balls in the uniform norm. Hilbert space case. General Sytaya theorem expressing the small balls probabilities in terms of the logarithm of Laplace transform. Zolotarev theorem for the case of regularly varying eigenvalues of covariance operator. Comparison results. Asymptotic behavior of Gaussian probabilities of small sets converging to non-zero point. Comparison theorem for the probabilities of small sets for two different mutually absolutely continuous Gaussian measures.

19. Open problems: Ten actual problems of the theory of Gaussian random functions and distributions are suggested. They should open to the interested reader a horizon for research in this field. Comments: Historical comments and references for additional sources for each section. Reference List: Approximately 400 references to Russian and foreign papers and books on Gaussian measures and random functions.

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