Lectures on Gaussian Processes

Lifshits, Mikhail

doi:10.1007/978-3-642-24939-6_1

Cited by 45 publications

(96 citation statements)

References 146 publications

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“…Hence, in view of (19), the Cameron-Martin space H X = RH X consists of absolutely continuous functions having square integrable derivative and vanishing at 0 and 1, which agrees with the well known description of this RKHS, see e.g. [17,Example 4.9]. Similarly to the previous example, the driftf from Theorem 2.9 should satisfyf ≥ f and f ′ −f ′ ,f ′ L 2 [0,1] ≥ 0.…”

Section: For Any Usupporting

confidence: 84%

“…We note in passing that comparable lower bounds to (1) follow also by [16,Theorem 1.1'] or [17,Theorem 7.3]. From the above, if further P 0,u ∈ (0, 1), then…”

Section: Introductionsupporting

confidence: 71%

“…We slightly abuse rigor here, because the space H X is a completion of the quotient M(T 1 )/ ker R, not a completion of M(T 1 ). For example, the book [17] goes through I * : M(T 1 ) → M(T 1 )/ ker R. However, we decided to keep this slightly ambiguous notation for the sake of clarity and simplicity and in view of the fact that the main results of this article are formulated for the case where R is injective.…”

Section: Resultsmentioning

confidence: 99%

“…As above, the operator J a extends to an isomorphism between H X and some subspace of ([a, b]) is a bit trickier. As in the previous example, by (17), it contains concave non-decreasing functions, but not all of them. In fact, it is easy to see from (17) that we must have f ′ + (a) = µ((a, b]) ≤ µ([a, b]) = f (a)/a ≥ 0; also every concave nondecreasing function with such property belongs to the image.…”

Section: For Anymentioning

confidence: 87%

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Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics

2020

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We study boundary non-crossing probabilitiesfor continuous centered Gaussian process X indexed by some arbitrary compact separable metric space T. We obtain both upper and lower bounds for P f,u . The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case P cf,u , c → +∞, which are two-term approximations (up to o(c)). The asymptotics are formulated in terms of the solutionf to the constrained optimization problem h HX → min, h ∈ H X , h ≥ f in the reproducing kernel Hilbert space H X of X. Several applications of the results are further presented.2010 Mathematics Subject Classification. 60G15; 60G70; 60F10.

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Section: For Any Usupporting

confidence: 84%

“…We note in passing that comparable lower bounds to (1) follow also by [16,Theorem 1.1'] or [17,Theorem 7.3]. From the above, if further P 0,u ∈ (0, 1), then…”

Section: Introductionsupporting

confidence: 71%

Section: Resultsmentioning

confidence: 99%

Section: For Anymentioning

confidence: 87%

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Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics

2020

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“…To be able to state about the sensitivity of the MOSUM test, we discuss a comparison study by defining a similar test using the integrated regression function under alternative with respect to univariate Gaussian white noise which is a random filed defined e.g., in Alexander and Pyke (1986;Pyke, 1983;Gaensler 1993;Lifshits, 2012). This statistic is actually the limit of that defined as the integral with respect to CUSUM process of the arrays of the observations.…”

Section: Introductionmentioning

confidence: 99%

Accessing the Appropriateness of a Spatial Regression Using Generalized (h1h2-)Slepian Random Field

Somayasa¹,

Safani²,

Ngkoimani³

et al. 2017

Journal of Mathematics and Statistics

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Abstract:In this study we derived asymptotic goodness-of-fit test (model check) for spatial regression where the critical region as well as the pvalue of the tests are approximated based on the distribution of a type of the integral functional of the generalized (h 1 h 2 -)-Slepian field and the setindexed Gaussian white noise. Such random fields are obtained as the limit process of the moving and the cumulative sums processes of the sequence of random matrices consist of independent and identically distributed random variables indexed by the points of a design constructed by means of a given continuous probability measure. Although the common approach in model diagnostic for regression is based on the functional of the residuals, in this study a new different idea is proposed by directly investigating the moving and the cumulative sums of the array of the observations. It is shown that these approaches are mathematically tractable and practically more applicable. Simulation study is conducted for investigating the finite sample size behavior of the tests. An application of the procedure to a mining data is also discussed, where from the perspective of geology and geophysics, polynomial model is reasonable and suitable for the data.

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