Upper and Lower Bounds for Ruin Probability

In this note we discuss upper and lower bound for the ruin probability in an insurance model with very heavy-tailed claims and interarrival times

Relationship between Extremal and Sum Processes Generated by the same Point Process

2000 Mathematics Subject Classification: Primary 60G51, secondary 60G70, 60F17.We discuss weak limit theorems for a uniformly negligible triangular array (u.n.t.a.) in Z = [0, ∞) × [0, ∞)^d as well as for the associated with it sum and extremal processes on an open subset S . The complement of S turns out to be the explosion area of the limit Poisson point process. In order to prove our criterion for weak convergence of the sum processes we introduce and study sum processes over explosion area. Finally we generalize the model of u.n.t.a. to random sample size processes

Nonlinear Normalization in Limit Theorems for Extremes

2000 Mathematics Subject Classification: 60G70, 60F05.It is well known that under linear normalization the maxima of iid random variables converges in distribution to one of the three types of max-stable laws: Frechet, Gumbel and Weibull. During the last two decades the first author and her collaborators worked out a limit theory for extremes and extremal processes under non-linear but monotone normalizing mappings. In this model there is only one type of max-stable distributions and all continuous and strictly increasing df's belong to it. In a recent paper on General max-stable laws, Sreehari points out two "confusing" results in Pancheva (1984). They concern the explicit form of a max-stable df with respect to a continuous one-parameter group of max-automorphisms, and domain of attraction conditions. In the present paper the first claim is answered by a detailed explanation of the explicit form, while for the second we give a revised proof. The rate of convergence is also discussed

Extreme value distributions and Renormalization Group

In the classical theorems of extreme value theory the limits of suitably rescaled maxima of sequences of independent, identically distributed random variables are studied. So far, only affine rescalings have been considered. We show, however, that more general rescalings are natural and lead to new limit distributions, apart from the Gumbel, Weibull, and Fr\'echet families. The problem is approached using the language of Renormalization Group transformations in the space of probability densities. The limit distributions are fixed points of the transformation and the study of the differential around them allows a local analysis of the domains of attraction and the computation of finite-size corrections.Comment: 16 pages, 5 figures. Final versio

High- and mid-latitude quasi-2-day waves observed simultaneouslyby four meteor radars during summer 2000

International audienceResults from the analysis of MLT wind measurements at Dixon (73.5°N, 80°E), Esrange (68°N, 21°E), Castle Eaton (UK) (53°N, 2°W), and Obninsk (55°N, 37°E) during summer 2000 are presented in this paper. Using S-transform or wavelet analysis, quasi-two-day waves (QTDWs) are shown to appear simultaneously at high- and mid-latitudes and reveal themselves as several bursts of wave activity. At first this activity is preceded by a 51?53h wave with S=3 observed mainly at mid-latitudes. After a short recess (or quiet time interval for about 10 days near day 205), we observe a regular sequence of three bursts, the strongest of them corresponding to a QTDW with a period of 47?48h and S=4 at mid-altitudes. We hypothesize that these three bursts may be the result of constructive and destructive interference between several spectral components: a 47?48h component with S=4; a 60-h component with S=3; and a 80-h component with S=2. The magnitudes of the lower (higher) zonal wave-number components increase (decrease) with increasing latitude. The S-transform or wavelet analysis indicates when these spectral components create the wave activity bursts and gives a range of zonal wave numbers for observed bursts from about 4 to about 2 for mid- and high-latitudes. The main spectral component at Dixon and Esrange latitudes is the 60-h oscillation with S=3. The zonal wave numbers and frequencies of the observed spectral components hint at the possible occurrence of the nonlinear interaction between the primary QTDWs and other planetary waves. Using a simple 3-D nonlinear numerical model, we attempt to simulate some of the observed features and to explain them as a consequence of the nonlinear interaction between the primary 47?48h and the 9?10day waves, and the resulting linear superposition of primary and secondary waves. In addition to the QTDW bursts, we also infer forcing of the 4-day wave with S=2 and the 6?7day wave with S=1, possibly arising from nonlinear decoupling of the 60-h wave with S=3. The starting mechanism for this decoupling is the Rossby wave instability (e.g. Baines, 1976). This result is consistent with the day-to-day wind variability during the observed QTDW events. An interesting feature of the final stage of the observed QTDW activity in summer 2000 is the occurrence of strong 4?5 day waves with S=3. Key words. Meteorology and atmospheric dynamics (middle atmosphere dynamics; waves and tides; general or miscellaneous

On the max-semistable limit of maxima of stationary sequences with missing values

Let [X(n)] be a stationary sequence with marginal distribution in the domain of attraction of a max-semistable distribution. This includes all distributions in the domain of attraction of any max-stable distribution and also other distributions like some integer-valued distributions with exponential type tails such as the Negative Binomial case. We consider the effect of missing values on the distribution of the maximum term. The pattern Of Occurrence of the missing values must be either iid or strongly mixing. We obtain the expression of the extremal index for the resulting sequence. The results generalize and extend the ones obtained for the max-stable domain of attraction. (c) 2008 Elsevier B.V. All rights reserved

Latitudinal wave coupling of the stratosphere and mesosphere during the major stratospheric warming in 2003/2004

The coupling of the dynamical regimes in the high- and low-latitude stratosphere and mesosphere during the major SSW in the Arctic winter of 2003/2004 has been studied. The UKMO zonal wind data were used to explore the latitudinal coupling in the stratosphere, while the coupling in the mesosphere was investigated by neutral wind measurements from eleven radars situated at high, high-middle and tropical latitudes. It was found that the inverse relationship between the variability of the zonal mean flows at high- and low-latitude stratosphere related to the SSW is produced by global-scale zonally symmetric waves. Their origin and other main features have been investigated in detail. Similar latitudinal dynamical coupling has been found for the mesosphere as well. Indirect evidence for the presence of zonally symmetric waves in the mesosphere has been found