The trivariate extremes based on a random sample size

In this paper, the class of the limit distribution functions (d.f.'s) of the trivariate extreme order statistics (upper-upper-upper (u-u-u), or upper-upper-lower (u-u-1)) with random sample size, which is assumed to be independent of all basic random variables (r.v.'s), is fully characterized. The necessary and sufficient conditions for the weak convergence (->(w)) of the distributions of the trinal vectors of the extremes with random sample sizes are obtained. Finally, an illustrative example is given.