Markov tail chains

The extremes of a univariate Markov chain with regulary varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper, we extend this fact to Markov chains with multivariate regularly varying marginal distribution in R^d. We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, it will be seen that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is Markovian as well. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event

Statistics for Tail Processes of Markov Chains

At high levels, the asymptotic distribution of a stationary, regularly varying Markov chain is conveniently given by its tail process. The latter takes the form of a geometric random walk, the increment distribution depending on the sign of the process at the current state and on the flow of time, either forward or backward. Estimation of the tail process provides a nonparametric approach to analyze extreme values. A duality between the distributions of the forward and backward increments provides additional information that can be exploited in the construction of more efficient estimators. The large-sample distribution of such estimators is derived via empirical process theory for cluster functionals. Their finite-sample performance is evaluated via Monte Carlo simulations involving copula-based Markov models and solutions to stochastic recurrence equations. The estimators are applied to stock price data to study the absence or presence of symmetries in the succession of large gains and losses

Use of stereo camera systems for assessment of rockfish abundance in untrawlable areas and for recording pollock behavior during midwater trawls

We describe the application of two types of stereo camera systems in fisheries research, including the design, calibration, analysis techniques, and precision of the data obtained with these systems. The first is a stereo video system deployed by using a quick-responding winch with a live feed to provide species- and size- composition data adequate to produce acoustically based biomass estimates of rockfish. This system was tested on the eastern Bering Sea slope where rockfish were measured. Rockfish sizes were similar to those sampled with a bottom trawl and the relative error in multiple measurements of the same rockfish in multiple still-frame images was small. Measurement errors of up to 5.5% were found on a calibration target of known size. The second system consisted of a pair of still-image digital cameras mounted inside a midwater trawl. Processing of the stereo images allowed fish length, fish orientation in relation to the camera platform, and relative distance of the fish to the trawl netting to be determined. The video system was useful for surveying fish in Alaska, but it could also be used broadly in other situations where it is difficult to obtain species-composition or size-composition information. Likewise, the still-image system could be used for fisheries research to obtain data on size, position, and orientation of fish

Tetra­ethyl­ammonium 4-hy­droxy­benzoate monohydrate

In the title compound, C8H20N+·C7H5O3 −·H2O, the carboxyl­ate group is slightly out of the plane of the parent benzene ring, the C—C—C—O torsion angles being 2.3 (2) and 2.0 (2)°. The carboxyl­ate group and the hy­droxy group form O—H⋯O hydrogen bonds, generating a head-to-tail chain along the b axis. Neighbouring hydrogen-bonded chains are linked by the water mol­ecule, generating two independent O—H⋯O donor hydrogen bonds. The carboxyl­ate group thus constructs a hydrogen-bonded host layer parallel to (10). The tetra­ethyl­ammonium cation is contained between these layers, forming a sandwich-like structure with an approximate inter­layer distance of 10.03 Å