Random Sample Sizes: Limit Theorems and Characterizations

Abstract

Random sample sizes naturally come up in such topics as sequential analysis, branching processes, damage models or rarefactions of point processes, and records as maxima, while their introduction in an applied model permits the user to select samples of varying sizes on different occasions. In the first group of examples, the random sample size is generated by the problem itself, hence the mathematician has no control over the dependence between the sample size and the underlying random variables. On the other hand, if one introduces the random sample size as an extension of a model (mainly for statistical inference), one can usually assume that it is independent of the underlying variables. Therefore, one has to be aware of the limitations of using a result developed under the assumption of the sample size’s independence of the main variables.