Transfer theorems concerning asymptotic expansions for the distribution functions of statistics based on samples with random sizes

In the paper, we discuss the transformation of the asymptotic expansion for the distribution of a statistic admitting Edgeworth expansion if the sample size is replaced by a random variable.We demonstrate that all those statistics that are regarded as asymptotically normal in the classical sense, become asymptotically Laplace or Student if the sample size is random. We especially separate the case where the Student distribution parameter ("the number degrees of freedom") is small. We show that the Student distribution with arbitrary "number of degrees of freedom "can be obtained as the limit when the sample size is random. We emphasize the possibility of using a family of Student distributions as a comfortable model with heavy tails since in this case many relations, in particular, a likelihood function, have the explicit form (unlike stable laws). Thus, the Laplace and Student distributions may be used as an asymptotic approximation in descriptive statistics being a convenient heavy-tailed alternative to stable laws. © 2018 Jangjeon Mathematical Society. All rights reserved.