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Regular variation, ℓ-max-stable laws, p-max-stable laws, Uniform metric, Total variation matric, Primary 60F05, 62E20, Secondary 62E15, 62G30,

The class of limit distribution functions (df's) of multivariate order statistics from independent and identical random vectors with random sample size is fully characterized. Two cases in this study are considered, the first case is when the random sample size is assumed to be independent of all the basic random vectors, and the second case is when the interrelation of the random size and the basic random vectors is not restricted.