## Lectures on Gaussian ProcessesGaussian processes can be viewed as a far-reaching infinite-dimensional extension of classical normal random variables. Their theory presents a powerful range of tools for probabilistic modelling in various academic and technical domains such as Statistics, Forecasting, Finance, Information Transmission, Machine Learning - to mention just a few. The objective of these Briefs is to present a quick and condensed treatment of the core theory that a reader must understand in order to make his own independent contributions. The primary intended readership are PhD/Masters students and researchers working in pure or applied mathematics. The first chapters introduce essentials of the classical theory of Gaussian processes and measures with the core notions of reproducing kernel, integral representation, isoperimetric property, large deviation principle. The brevity being a priority for teaching and learning purposes, certain technical details and proofs are omitted. The later chapters touch important recent issues not sufficiently reflected in the literature, such as small deviations, expansions, and quantization of processes. In university teaching, one can build a one-semester advanced course upon these Briefs. |

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assumption asymptotic Banach space base e j Borell Brownian bridge Brownian sheet centered Gaussian process centered Gaussian random centered Gaussian vector compact concavity consider convex sets corresponding covariance function covariance operator covering numbers deﬁnition denote density dispersion ellipsoid dual space Euclidean exists expansion ﬁnd ﬁnite ﬁrst FLIL follows fractional Brownian motion functional f Gaussian distribution Gaussian measure Gaussian processes GC-set Hilbert space independent integral representation isoperimetric inequality iterated logarithm Large Deviation Principle Lebesgue measure Lemma Lévy Lifshits lim sup linear functionals linear space Math metric entropy norm normal distribution obtain proved quantization random element random process random vector Recall sample paths satisﬁes scalar product sequence small ball small deviation function small deviation probabilities standard Gaussian standard normal stationary processes sufﬁciently sup X(t symmetric convex sets tGT tGT theory true unit ball white noise Wiener process yields