The Annals of probability, ISSN 0091-1798, 2009, Volume 37, Issue 5, pp. 2042 - 2065

Let $W_{i}, i \in \mathbb{N}$ , be independent copies of a zero-mean Gaussian process ${W(t), t \in \mathbb{R}^{d}}$ with stationary increments and variance...

Brownian motion | Gaussian distributions | Covariance | Stochastic processes | Eigenfunctions | Laplace transformation | Mathematical functions | Mathematical maxima | Cylinders | Perceptron convergence procedure | Stationary max-stable processes | Extremes | Gaussian processes | Poisson point processes | EXTREME VALUES | MULTIVARIATE | STATISTICS & PROBABILITY | extremes | Mathematics - Probability | 60G70 | 60G15

Brownian motion | Gaussian distributions | Covariance | Stochastic processes | Eigenfunctions | Laplace transformation | Mathematical functions | Mathematical maxima | Cylinders | Perceptron convergence procedure | Stationary max-stable processes | Extremes | Gaussian processes | Poisson point processes | EXTREME VALUES | MULTIVARIATE | STATISTICS & PROBABILITY | extremes | Mathematics - Probability | 60G70 | 60G15

Journal Article

Journal of theoretical probability, ISSN 1572-9230, 2019

We derive exact tail asymptotics of sojourn time above the level $u\geq 0$ $$
\mathbb{P}\left(v(u)\int_0^T \mathbb{I}(X(t)-ct>u)d t>x\right), \quad x\geq 0
$$...

Mathematics - Probability

Mathematics - Probability

Journal Article

The Annals of applied probability, ISSN 1050-5164, 2/2003, Volume 13, Issue 1, pp. 37 - 53

We study the asymptotics for the maximum on a random time interval of a random walk with a long-tailed distribution of its increments and negative drift. We...

Integers | Logical proofs | Random walk | Stochastic processes | Random variables | Indicator functions | Stopping distances | Distribution functions | Ruin probability | Long-tailed distribution | Subexponential distribution | DISTRIBUTIONS | STATISTICS & PROBABILITY | ruin probability | long-tailed distribution | subexponential distribution | 60K30 | 60G70 | 60K25

Integers | Logical proofs | Random walk | Stochastic processes | Random variables | Indicator functions | Stopping distances | Distribution functions | Ruin probability | Long-tailed distribution | Subexponential distribution | DISTRIBUTIONS | STATISTICS & PROBABILITY | ruin probability | long-tailed distribution | subexponential distribution | 60K30 | 60G70 | 60K25

Journal Article

Journal of applied probability, ISSN 0021-9002, 06/2007, Volume 44, Issue 2, pp. 428 - 443

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram...

Insurance risk process | Reflected Lévy process | Integrated exponential Lévy process | Passage problem | Ruin | insurance risk process | 60K05 | 60G70 | passage problem | 60K15 | ruin | 91B30 | integrated exponential Lévy process | 60J55

Insurance risk process | Reflected Lévy process | Integrated exponential Lévy process | Passage problem | Ruin | insurance risk process | 60K05 | 60G70 | passage problem | 60K15 | ruin | 91B30 | integrated exponential Lévy process | 60J55

Journal Article

The Annals of statistics, ISSN 0090-5364, 4/2015, Volume 43, Issue 2, pp. 903 - 934

The estimation of the extremal dependence structure is spoiled by the impact of the bias, which increases with the number of observations used for the...

Bias correction | Threshold choice | Tail dependence | Multivariate extreme value theory | PROBABILITY | bias correction | MODELS | STATISTICS | ESTIMATORS | STATISTICS & PROBABILITY | INDEX | tail dependence | PICKANDS DEPENDENCE | threshold choice | Statistics | Statistics Theory | Mathematics | 62G05 | 62G20 | 60G70 | 62G32 | 60F05

Bias correction | Threshold choice | Tail dependence | Multivariate extreme value theory | PROBABILITY | bias correction | MODELS | STATISTICS | ESTIMATORS | STATISTICS & PROBABILITY | INDEX | tail dependence | PICKANDS DEPENDENCE | threshold choice | Statistics | Statistics Theory | Mathematics | 62G05 | 62G20 | 60G70 | 62G32 | 60F05

Journal Article

Journal of applied probability, ISSN 0021-9002, 03/2015, Volume 52, Issue 1, pp. 68 - 81

Let
W
= {
W
n
: n ∈ N} be a sequence of random vectors in R
d
, d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of
W
, that is,...

Large deviation theory | Extrema of stochastic process | 60G70 | 60F10 | large deviation theory

Large deviation theory | Extrema of stochastic process | 60G70 | 60F10 | large deviation theory

Journal Article

Journal of applied probability, ISSN 0021-9002, 09/2014, Volume 51, Issue 3, pp. 713 - 726

Define a γ-reflected process W
γ(t) = Y
H
(t) - γinf
s∈[0,t]
Y
H
(s), t ≥ 0, with input process {Y
H
(t), t ≥ 0}, which is a fractional Brownian motion with...

Passage time | Piterbarg constant | γ-reflected process | Risk process with tax | Workload process | Fractional Brownian motion | Pickands constant | Gaussian approximation | workload process | 60G70 | fractional Brownian motion | passage time | risk process with tax | 60G15

Passage time | Piterbarg constant | γ-reflected process | Risk process with tax | Workload process | Fractional Brownian motion | Pickands constant | Gaussian approximation | workload process | 60G70 | fractional Brownian motion | passage time | risk process with tax | 60G15

Journal Article

The Annals of statistics, ISSN 0090-5364, 10/2009, Volume 37, Issue 5B, pp. 2953 - 2989

Consider a random sample from a bivariate distribution function F in the max-domain of attraction of an extreme-value distribution function G. This G is...

Average linear density | Atoms | Sine function | Atomic spectra | Cauchy Lorentz distribution | Spectral index | Parametric models | Estimators | Logistics | Distribution functions | Moment constraint | Multivariate extremes | Functional central limit theorem | Tail dependence | National Health and Nutrition Examination Survey | Nonparametric maximum likelihood estimator | Local empirical process | local empirical process | multivariate extremes | moment constraint | nonparametric maximum likelihood estimator | STATISTICS & PROBABILITY | tail dependence | 62G05 | 60G70 | 62G30 | 62G32 | 60F17 | 60F05

Average linear density | Atoms | Sine function | Atomic spectra | Cauchy Lorentz distribution | Spectral index | Parametric models | Estimators | Logistics | Distribution functions | Moment constraint | Multivariate extremes | Functional central limit theorem | Tail dependence | National Health and Nutrition Examination Survey | Nonparametric maximum likelihood estimator | Local empirical process | local empirical process | multivariate extremes | moment constraint | nonparametric maximum likelihood estimator | STATISTICS & PROBABILITY | tail dependence | 62G05 | 60G70 | 62G30 | 62G32 | 60F17 | 60F05

Journal Article

The Annals of applied probability, ISSN 1050-5164, 8/2014, Volume 24, Issue 4, pp. 1446 - 1481

We study the statistics of the extremes of a discrete Gaussian field with logarithmic correlations at the level of the Gibbs measure. The model is defined on...

Brownian motion | Statistical variance | Spin glasses | Statistical theories | Correlations | Mathematical independent variables | Random variables | Statistics | Free energy | Probabilities | Gibbs measure | Log-correlated Gaussian fields | Tree approximation | Poisson Dirichlet variable | MAXIMUM | NONHIERARCHICAL VERSION | Poisson-Dirichlet variable | GHIRLANDA-GUERRA IDENTITIES | HIERARCHIES | STATISTICS & PROBABILITY | RANDOM ENERGY-MODEL | ULTRAMETRICITY | tree approximation | spin glasses | Probability | Condensed Matter | Disordered Systems and Neural Networks | Mathematics | Physics | 82B26 | Poisson–Dirichlet variable | 82B44 | 60G70 | 60G15 | 60F05

Brownian motion | Statistical variance | Spin glasses | Statistical theories | Correlations | Mathematical independent variables | Random variables | Statistics | Free energy | Probabilities | Gibbs measure | Log-correlated Gaussian fields | Tree approximation | Poisson Dirichlet variable | MAXIMUM | NONHIERARCHICAL VERSION | Poisson-Dirichlet variable | GHIRLANDA-GUERRA IDENTITIES | HIERARCHIES | STATISTICS & PROBABILITY | RANDOM ENERGY-MODEL | ULTRAMETRICITY | tree approximation | spin glasses | Probability | Condensed Matter | Disordered Systems and Neural Networks | Mathematics | Physics | 82B26 | Poisson–Dirichlet variable | 82B44 | 60G70 | 60G15 | 60F05

Journal Article

Advances in applied probability, ISSN 0001-8678, 03/2013, Volume 45, Issue 1, pp. 186 - 213

An asymptotic model for the extreme behavior of certain Markov chains is the ‘tail chain’. Generally taking the form of a multiplicative random walk, it is...

General Applied Probability | Markov chain | Transition kernel | Tail chain | Multivariate regular variation | Heavy tail | Extreme values | tail chain | heavy tail | multivariate regular variation | 60G70 | 62P05 | 60J05 | transition kernel

General Applied Probability | Markov chain | Transition kernel | Tail chain | Multivariate regular variation | Heavy tail | Extreme values | tail chain | heavy tail | multivariate regular variation | 60G70 | 62P05 | 60J05 | transition kernel

Journal Article

The Annals of statistics, ISSN 0090-5364, 2/2006, Volume 34, Issue 1, pp. 146 - 168

The aim of this paper is to provide models for spatial extremes in the case of stationarity. The spatial dependence at extreme levels of a stationary process...

Two dimensional modeling | Gaussian distributions | Stochastic processes | Stationary processes | Spectral energy distribution | Spatial models | Density | Spatial Statistics | Estimators | Consistent estimators | Distribution functions | Maxstable processes | Semiparametric estimation | Spatial extremes | Spatial tail dependence | Multivariate extremes | Extreme-value theory | spatial tail dependence | serniparametric estimation | SAMPLE | multivariate extremes | spatial extremes | STOCHASTIC-PROCESSES | STATISTICS & PROBABILITY | extreme-value theory | maxstable processes | max-stable processes | semiparametric estimation | 60G10 | 62E20 | 60G70 | 62G32 | 62H11 | 62M40

Two dimensional modeling | Gaussian distributions | Stochastic processes | Stationary processes | Spectral energy distribution | Spatial models | Density | Spatial Statistics | Estimators | Consistent estimators | Distribution functions | Maxstable processes | Semiparametric estimation | Spatial extremes | Spatial tail dependence | Multivariate extremes | Extreme-value theory | spatial tail dependence | serniparametric estimation | SAMPLE | multivariate extremes | spatial extremes | STOCHASTIC-PROCESSES | STATISTICS & PROBABILITY | extreme-value theory | maxstable processes | max-stable processes | semiparametric estimation | 60G10 | 62E20 | 60G70 | 62G32 | 62H11 | 62M40

Journal Article

The Annals of applied probability, ISSN 1050-5164, 4/2007, Volume 17, Issue 2, pp. 537 - 571

Models based on assumptions of multivariate regular variation and hidden regular variation provide ways to describe a broad range of extremal dependence...

Statistical models | Radon | Standardization | Mathematical vectors | Mathematical functions | Mathematics | Random variables | Asymptotic value | Perceptron convergence procedure | Distribution functions | Heavy tails | Hidden regular variation | Asymptotic independence | Regular variation | Coefficient of tail dependence | Conditional models | heavy tails | coefficient of tail dependence | INDEPENDENCE | BEHAVIOR | MULTIVARIATE | STATISTICS & PROBABILITY | conditional models | hidden regular variation | regular variation | asymptotic independence | 60G70 | 62G32

Statistical models | Radon | Standardization | Mathematical vectors | Mathematical functions | Mathematics | Random variables | Asymptotic value | Perceptron convergence procedure | Distribution functions | Heavy tails | Hidden regular variation | Asymptotic independence | Regular variation | Coefficient of tail dependence | Conditional models | heavy tails | coefficient of tail dependence | INDEPENDENCE | BEHAVIOR | MULTIVARIATE | STATISTICS & PROBABILITY | conditional models | hidden regular variation | regular variation | asymptotic independence | 60G70 | 62G32

Journal Article

Journal of applied probability, ISSN 0021-9002, 12/2012, Volume 49, Issue 4, pp. 1106 - 1118

Let {X
n
(t), t∈[0,∞)}, n∈ℕ, be standard stationary Gaussian processes. The limit distribution of
t∈[0,T(n)]|X
n
(t)| is established as r
n
(t), the...

Stationary Gaussian process | Pickands' constant | Strong dependence | Berman's condition | Limit theorem | Mathematics - Probability | strong dependence | 60G70 | limit theorem | 60G15

Stationary Gaussian process | Pickands' constant | Strong dependence | Berman's condition | Limit theorem | Mathematics - Probability | strong dependence | 60G70 | limit theorem | 60G15

Journal Article

The Annals of applied probability, ISSN 1050-5164, 8/2008, Volume 18, Issue 4, pp. 1351 - 1378

Let $(X_{n}\colon n\geq 0)$ be a sequence of i.i.d. r.v.'s with negative mean. Set S₀ = 0 and define $S_{n}=X_{1}+\cdots +X_{n}$. We propose an importance...

Estimate reliability | Approximation | Algorithms | Random walk | Insurance risk | Random variables | Traffic estimation | Sampling distributions | Estimators | Probabilities | Single-server queue | Change-of-measure | Random walks | Rare-event simulation | State-dependent importance sampling | Heavy-tails | Lyapunov bounds | MONTE-CARLO | random walks | rare-event simulation | change-of-measure | STATISTICS & PROBABILITY | heavy-tails | single-server queue | state-dependent importance sampling | Mathematics - Probability | 60J20 | 68W40 | 60G50 | 60G70 | 60J05

Estimate reliability | Approximation | Algorithms | Random walk | Insurance risk | Random variables | Traffic estimation | Sampling distributions | Estimators | Probabilities | Single-server queue | Change-of-measure | Random walks | Rare-event simulation | State-dependent importance sampling | Heavy-tails | Lyapunov bounds | MONTE-CARLO | random walks | rare-event simulation | change-of-measure | STATISTICS & PROBABILITY | heavy-tails | single-server queue | state-dependent importance sampling | Mathematics - Probability | 60J20 | 68W40 | 60G50 | 60G70 | 60J05

Journal Article

Advances in applied probability, ISSN 0001-8678, 03/2013, Volume 45, Issue 1, pp. 139 - 163

Multivariate regular variation plays a role in assessing tail risk in diverse applications such as finance, telecommunications, insurance, and environmental...

General Applied Probability | Risk set | Spectral measure | Vague Convergence | Asymptotic independence | Regular variation | 60F99 | risk set | 60G70 | 62G32 | vague convergence | regular variation | spectral measure | asymptotic independence

General Applied Probability | Risk set | Spectral measure | Vague Convergence | Asymptotic independence | Regular variation | 60F99 | risk set | 60G70 | 62G32 | vague convergence | regular variation | spectral measure | asymptotic independence

Journal Article

Advances in applied probability, ISSN 0001-8678, 12/2012, Volume 44, Issue 4, pp. 1142 - 1172

Let {X
i} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C
1, C
2,…) be a...

General Applied Probability | Randomly weighted sum | Intermediate regular variation | Regular variation | Heavy tail | Breiman's theorem | Randomly stopped sum | heavy tail | 60F10 | 60G50 | 60J80 | randomly stopped sum | 60G70 | intermediate regular variation | regular variation

General Applied Probability | Randomly weighted sum | Intermediate regular variation | Regular variation | Heavy tail | Breiman's theorem | Randomly stopped sum | heavy tail | 60F10 | 60G50 | 60J80 | randomly stopped sum | 60G70 | intermediate regular variation | regular variation

Journal Article

Probability theory and related fields, ISSN 1432-2064, 2012, Volume 157, Issue 1-2, pp. 405 - 451

It has been conjectured since the work of Lalley and Sellke (Ann. Probab., 15, 1052–1061, 1987) that branching Brownian motion seen from its tip (e.g. from its...

Mathematical and Computational Biology | Statistics for Business/Economics/Mathematical Finance/Insurance | Theoretical, Mathematical and Computational Physics | Operations Research/Decision Theory | 60J80 | Probability Theory and Stochastic Processes | 60G70 | Mathematics | Quantitative Finance | KPP EQUATION | STATISTICS & PROBABILITY | Computer science | Studies | Probability distribution | Markov analysis | Poisson distribution | Brownian motion | Reproduction | Random walk | Probability theory | Atomic structure | Statistics | Invariants | Decoration | Probability

Mathematical and Computational Biology | Statistics for Business/Economics/Mathematical Finance/Insurance | Theoretical, Mathematical and Computational Physics | Operations Research/Decision Theory | 60J80 | Probability Theory and Stochastic Processes | 60G70 | Mathematics | Quantitative Finance | KPP EQUATION | STATISTICS & PROBABILITY | Computer science | Studies | Probability distribution | Markov analysis | Poisson distribution | Brownian motion | Reproduction | Random walk | Probability theory | Atomic structure | Statistics | Invariants | Decoration | Probability

Journal Article

Journal of applied probability, ISSN 0021-9002, 09/2015, Volume 52, Issue 3, pp. 688 - 702

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal...

Research Papers | normal approximation | generalized Pickands' constant | Parisian ruin time | 60G70 | Parisian ruin probability | fractional Brownian motion | self-similar Gaussian process | 60G15

Research Papers | normal approximation | generalized Pickands' constant | Parisian ruin time | 60G70 | Parisian ruin probability | fractional Brownian motion | self-similar Gaussian process | 60G15

Journal Article

The Annals of probability, ISSN 0091-1798, 5/2012, Volume 40, Issue 3, pp. 1069 - 1104

This paper develops asymptotic approximations of P(∫ T e f(t) dt > b) as b → ∞ for a homogeneous smooth Gaussian random field, f, living on a compact...

Approximation | Covariance | Mathematical integrals | Logical proofs | Log integral function | Eigenvalues | Finite sums | Jordan matrices | Matrices | Random variables | Extremes | Gaussian random field | BROWNIAN-MOTION | COUNTS | MAXIMUM | VALUE-AT-RISK | INEQUALITY | TIME-SERIES | STATISTICS & PROBABILITY | REGRESSION-MODEL | extremes | RANDOM-VARIABLES | SUMS | Mathematics - Probability | 60G70 | 60F10

Approximation | Covariance | Mathematical integrals | Logical proofs | Log integral function | Eigenvalues | Finite sums | Jordan matrices | Matrices | Random variables | Extremes | Gaussian random field | BROWNIAN-MOTION | COUNTS | MAXIMUM | VALUE-AT-RISK | INEQUALITY | TIME-SERIES | STATISTICS & PROBABILITY | REGRESSION-MODEL | extremes | RANDOM-VARIABLES | SUMS | Mathematics - Probability | 60G70 | 60F10

Journal Article

Probability theory and related fields, ISSN 1432-2064, 2018, Volume 174, Issue 1-2, pp. 501 - 551

We consider Metropolis dynamics of the Random Energy Model. We prove that the classical two-time correlation function that allows one to establish aging...

Spin glasses | Statistics for Business, Management, Economics, Finance, Insurance | Clock process | Mathematical and Computational Biology | 82C44 | Theoretical, Mathematical and Computational Physics | Probability Theory and Stochastic Processes | 60G70 | Mathematics | Quantitative Finance | Metropolis dynamics | Random environments | Operations Research/Decision Theory | Random dynamics | Aging | 60K35 | Lévy processes | Levy processes | STATISTICS & PROBABILITY | Analysis | Correlation | Asymptotic properties | Ice | Time correlation functions | Topology | Rescaling | Continuity (mathematics) | Convergence | Distribution functions

Spin glasses | Statistics for Business, Management, Economics, Finance, Insurance | Clock process | Mathematical and Computational Biology | 82C44 | Theoretical, Mathematical and Computational Physics | Probability Theory and Stochastic Processes | 60G70 | Mathematics | Quantitative Finance | Metropolis dynamics | Random environments | Operations Research/Decision Theory | Random dynamics | Aging | 60K35 | Lévy processes | Levy processes | STATISTICS & PROBABILITY | Analysis | Correlation | Asymptotic properties | Ice | Time correlation functions | Topology | Rescaling | Continuity (mathematics) | Convergence | Distribution functions

Journal Article

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