How Exceptional is the Extremal Kendall and Kendall-Type Convolution

AbstractThis paper deals with the generalized convolutions connected with the Williamson transform and the maximum operation. We focus on such convolutions which can define transition probabilities of renewal processes. They should be monotonic since the described time or destruction does not go back, it should admit existence of a distribution with a lack of memory property because the analog of the Poisson process shall exist. Another valuable property is the simplicity of calculating and inverting the corresponding generalized characteristic function (in particular Williamson transform) so that the technique of generalized characteristic function can be used in description of our processes. The convex linear combination property (the generalized convolution of two point measures is the convex combination of several fixed measures), or representability (which means that the generalized convolution can be easily written in the language of independent random variables)—they also facilitate the modeling of real processes in that language. We describe examples of generalized convolutions having the required properties ranging from the maximum convolution and its simplest generalization—the Kendall convolution (associated with the Williamson transform), up to the most complicated here—Kingman convolution. It is novel approach to apply in the extreme value theory. Stochastic representation of the Kucharczak-Urbanik in the order statistics terms is proved, which open new paths to investigate Archimedean copulas. This paper open the door to solve an old open problem of the relationship between copulas and generalized convolutions mentioned by B. Schweizer and A. Sklar in 1983. This indicates the path of further research towards extremes and dependency modelling.</jats:p

Teeth of the red fox Vulpes vulpes (L., 1758) as a bioindicator in studies on fluoride pollution

An examination was made of fluoride content in the mandibular first molars of the permanent teeth of the red fox Vulpes vulpes living in north-west (NW) Poland. The teeth were first dried to a constant weight at 105°C and then ashed. Fluorides were determined potentiometrically, and their concentrations were expressed in dry weight (DW) and ash. The results were used to perform an indirect estimation of fluoride pollution in the examined region of Poland. The collected specimens (n = 35) were classified into one of the three age categories: immature (im, 6–12 months), subadult (subad, from 12 to 20 months) and adult (ad, >20 months). The mean concentrations (geometric mean) of fluoride were similar in the im and subad groups (230 and 296 mg/kg DW and 297 and 385 mg/kg ash, respectively), and significantly smaller than in the ad group (504 and 654 mg/kg, respectively, in DW and ash). Basing on other reports that the ∼400 mg/kg DW concentration of fluoride in bones in the long-lived wild mammals generally reflects the geochemical background, it was found that 57% of the foxes in NW Poland exceeded this value by 9% to 170%. This indirectly reflects a moderate fluoride contamination in the tested region