On Piterbarg theorem for maxima of stationary Gaussian sequences

Limit distributions of maxima of dependent Gaussian sequence are different according to the convergence rate of their correlations. For three different conditions on convergence rate of the correlations, in this paper, we establish the Piterbarg theorem for maxima of stationary Gaussian sequence

The asymptotic distribution of maxima of stationary random sequences under random replacing

In this paper, we investigated the effect on extreme of random replacing for a stationary sequence satisfying a type of long dependent condition and a local dependent condition, and derived the joint asymptotic distribution of maximum from the stationary sequence and the maximum from the random replacing sequence. We also provided several applications for our main results.Comment:

Berman's inequality under random scaling

Berman's inequality is the key for establishing asymptotic properties of maxima of Gaussian random sequences and supremum of Gaussian random fields. This contribution shows that, asymptotically an extended version of Berman's inequality can be established for randomly scaled Gaussian random vectors. Two applications presented in this paper demonstrate the use of Berman's inequality under random scaling

Piterbarg's max-discretization theorem for stationary vector Gaussian processes observed on different grids

In this paper, we derive Piterbarg's max-discretization theorem for two different grids considering centred stationary vector Gaussian processes. So far in the literature results in this direction have been derived for the joint distribution of the maximum of Gaussian processes over and over a grid . In this paper, we extend the recent findings by considering additionally the maximum over another grid . We derive the joint limiting distribution of maximum of stationary Gaussian vector processes for different choices of such grids by letting . As a by-product we find that the joint limiting distribution of the maximum over different grids, which we refer to as the Piterbarg distribution, is in the case of weakly dependent Gaussian processes a max-stable distribution

On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields

In this paper, we consider the extreme behavior of a Gaussian random field $f(t)$ living on a compact set $T$. In particular, we are interested in tail events associated with the integral $\int_Te^{f(t)}\,dt$. We construct a (non-Gaussian) random field whose distribution can be explicitly stated. This field approximates the conditional Gaussian random field $f$ (given that $\int_Te^{f(t)}\,dt$ exceeds a large value) in total variation. Based on this approximation, we show that the tail event of $\int_Te^{f(t)}\,dt$ is asymptotically equivalent to the tail event of $\sup_T\gamma(t)$ where $\gamma(t)$ is a Gaussian process and it is an affine function of $f(t)$ and its derivative field. In addition to the asymptotic description of the conditional field, we construct an efficient Monte Carlo estimator that runs in polynomial time of $\log b$ to compute the probability $P(\int_Te^{f(t)}\,dt>b)$ with a prescribed relative accuracy.Comment: Published in at http://dx.doi.org/10.1214/13-AAP960 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

State space modelling of extreme values with particle filters

State space models are a flexible class of Bayesian model that can be used to smoothly capture non-stationarity. Observations are assumed independent given a latent state process so that their distribution can change gradually over time. Sequential Monte Carlo methods known as particle filters provide an approach to inference for such models whereby observations are added to the fit sequentially. Though originally developed for on-line inference, particle filters, along with related particle smoothers, often provide the best approach for off-line inference. This thesis develops new results for particle filtering and in particular develops a new particle smoother that has a computational complexity that is linear in the number of Monte Carlo samples. This compares favourably with the quadratic complexity of most of its competitors resulting in greater accuracy within a given time frame. The statistical analysis of extremes is important in many fields where the largest or smallest values have the biggest effect. Accurate assessments of the likelihood of extreme events are crucial to judging how severe they could be. While the extreme values of a stationary time series are well understood, datasets of extremes often contain varying degrees of non-stationarity. How best to extend standard extreme value models to account for non-stationary series is a topic of ongoing research. The thesis develops inference methods for extreme values of univariate and multivariate non-stationary processes using state space models fitted using particle methods. Though this approach has been considered previously in the univariate case, we identify problems with the existing method and provide solutions and extensions to it. The application of the methodology is illustrated through the analysis of a series of world class athletics running times, extreme temperatures at a site in the Antarctic, and sea-level extremes on the east coast of England

Advanced methods for analysing and modelling multivariate palaeoclimatic time series

The separation of natural and anthropogenically caused climatic changes is an important task of contemporary climate research. For this purpose, a detailed knowledge of the natural variability of the climate during warm stages is a necessary prerequisite. Beside model simulations and historical documents, this knowledge is mostly derived from analyses of so-called climatic proxy data like tree rings or sediment as well as ice cores. In order to be able to appropriately interpret such sources of palaeoclimatic information, suitable approaches of statistical modelling as well as methods of time series analysis are necessary, which are applicable to short, noisy, and non-stationary uni- and multivariate data sets. Correlations between different climatic proxy data within one or more climatological archives contain significant information about the climatic change on longer time scales. Based on an appropriate statistical decomposition of such multivariate time series, one may estimate dimensions in terms of the number of significant, linear independent components of the considered data set. In the presented work, a corresponding approach is introduced, critically discussed, and extended with respect to the analysis of palaeoclimatic time series. Temporal variations of the resulting measures allow to derive information about climatic changes ...thesi