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Several authors are interested in the study of limiting distributions for large symmetrical random matrices. Our approach of studying limiting distributions is different and is related to the new concept of variance functions of Cauchy–Stieltjes Kernel (CSK) families of probabilities. By means of the variance functions machinery, some novel limiting probability measures are provided involving the free multiplicative law of large numbers, using both free and Boolean (additive and multiplicative) convolutions. Several examples of these limiting probability measures are given in the context of free probability. [ABSTRACT FROM AUTHOR]

A notion of generalized (two-parameterized) t-transformation of free convolution, also called (t = (a , b)) -deformed free convolution, is introduced for a ∈ R and b > 0 . In this article, some results of t -deformed free convolution are given within the theory of Cauchy-Stieltjes Kernel (CSK) families. The variance function is a fundamental concept in CSK families. An expression is provided for the variance function under t -deformed free convolution power. In addition, through the use of the variance function, an approximation is provided for members of the t -deformed free Gaussian CSK family and members of the t -deformed free Poisson CSK family respectively. Furthermore, by involving the free multiplicative convolution, a new limit theorem is provided with respect to t -deformed free convolution. [ABSTRACT FROM AUTHOR]

One of the most used techniques for determining animal abundance is distance sampling. The distance sampling framework depends on the idea of a detection function, and a number of options have been suggested. In this paper, we provide a new flexible parametric model based on the Burr XII distribution. To be more specific, we use the survival function of the Burr XII distribution for novel purposes in this context. The proposed model is appealing because it meets all of the requirements for a reliable detection function model, such as being monotonically decreasing and having a shoulder at the origin. It also has the features of having various asymmetric properties and a heavy right tail, which are rare properties in this setting. In the first part, we provide its key characteristics, such as shapes and moments. Then, the inferential aspect of the model is investigated. The maximum likelihood estimation method is used to estimate the parameters in a data-fitting scenario. The estimates of population abundance are derived and compared with some existing parametric estimates. A simulation is run to assess how well the resulting estimates perform in comparison to other widely applied estimates from the literature. The model is then tested using two real-world data sets. Based on the famous goodness-of-fit statistics, we show that it is preferable to some of the well-established models. [ABSTRACT FROM AUTHOR]