Norms of Gaussian sample functions

Cirel'son, B. S.; Ибрагимов, И. А.; Sudakov, V. N.

doi:10.1007/bfb0077482

Cited by 151 publications

(123 citation statements)

References 5 publications

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“…The extremes of Gaussian random field have been extensively studied in literature, with special focus on approximations and bounds for the suprema (Piterbarg (1996), Sun (1993), Azais and Wschebor (2008), Borell (1975), Tsirelson, Ibragimov, and Sudakov (1976)). Also, there are closed form results on the excursion set A u such as Adler and Taylor (2007).…”

Section: Blanchet Liu and Yangmentioning

confidence: 99%

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Monte Carlo for large credit portfolios with potentially high correlations

Blanchet

Liu

Yang

2010

Proceedings of the 2010 Winter Simulation Conference

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In this paper we develop efficient Monte Carlo methods for large credit portfolios. We assume the default indicators admit a Gaussian copula. Therefore, we are able to embed the default correlations into a continuous Gaussian random field, which is capable of incorporating an infinite size portfolio and potentially highly correlated defaults. We are particularly interested in estimating the expectations, such as the expected number of defaults given that there is at least one default and the expected loss given at least one default. All these quantities turn out to be closely related to the geometric structure of the random field. We will heavily employ random field techniques to construct importance sampling based estimators and provide rigorous efficiency analysis.

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Section: Blanchet Liu and Yangmentioning

confidence: 99%

“…The first one is known as B-TIS ineuqality (Borell (1975), Tsirelson, Ibragimov, and Sudakov (1976)). …”

Section: Monte Carlo For Fields With Finite Expansion and Efficiency mentioning

confidence: 99%

Monte Carlo for large credit portfolios with potentially high correlations

Blanchet

Liu

Yang

2010

Proceedings of the 2010 Winter Simulation Conference

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“…Lemma 3 (Cirelson, Ibragimov and Sudakov (1976)) Let D be a subset of R. Let (η t ) t∈D be a centered Gaussian process. If E (sup t∈D η t ) ≤ N and sup t∈D V (η t ) ≤ V then, for any x > 0, we have…”

Section: Proof Of Lemma 1 Using the Inequalities Supmentioning

confidence: 99%

On adaptive wavelet estimation of a quadratic functional from a deconvolution problem

Chesneau

2009

Ann Inst Stat Math

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“…This estimate allows us, in the second part of this section, to describe a mechanism of a stable evolution. Here we use some biological arguments fundamental probabilistic tools (concentration inequalities, see [11,57,58,10] and also ideas of the recently developed theory of the phase transitions in hard combinatorial problems [77,13,14]). …”

Section: Organization Of the Paper And Main Mathematical Toolsmentioning

confidence: 99%

“…We apply here a powerful tool from the probability theory : concentration inequalities [11,10,57,58] and an idea of the phase transitions in hard combinatorial problems [77,13,14].…”

Section: Let Us Denotementioning

confidence: 99%

Instability, complexity, and evolution

Vakulenko

Grigoriev

2009

J Math Sci

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In this paper we consider a new class of random dynamical systems which contains in particular neural networks and complicated circuits. For these systems we consider the viability problem: we suppose that the system survives only if the system state is in a prescribed domain Π of a phase space. The approach developed here is based on some fundamental ideas proposed by A. Kolmogorov, R.Thom, M. Gromov, L.Valiant, L. Van Valen and others.Under some conditions it is shown that almost all systems from this class with fixed parameters are unstable in the following sense: the probability P t to leave Π within time interval [0, t] tends to 1 as t → ∞. However, if it is allowed to change these parameters sometimes ("evolutionary" case), then possibly that P t < 1 − δ < 1 for all t. ("stable ebolution"). Furthermore we study the properties of such stable evolution assuming that the system parameters are coded by a dicsrete code. This allows us to apply the complexity theory, coding, algorithms etc. Evolution is a Markov process of this code modification.Under some conditions we show that the stable evolution of unstable systems possesses such general fundamental property: the relative Kolmogorov complexity of the code cannot be bounded by a constant as time t → ∞.For circuit models we define complexity characteristics of these circuits. We find that these complexities also have a tendency to increase during stable evolution. We give concrete examples of stable evolution.

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Norms of Gaussian sample functions

Cited by 151 publications

References 5 publications

Monte Carlo for large credit portfolios with potentially high correlations

Monte Carlo for large credit portfolios with potentially high correlations

On adaptive wavelet estimation of a quadratic functional from a deconvolution problem

Instability, complexity, and evolution

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