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Most of the observable phenomena in the empirical sciences are of a multivariate nature. In financial studies, assets in stock markets are observed simultaneously and their joint development is analyzed to better understand general tendencies and to track indices. In medicine recorded observations of subjects in different locations are the basis of reliable diagnoses and medication. In quantitative marketing consumer preferences are collected in order to construct models of consumer behavior. The underlying theoretical structure of these and many other quantitative studies of applied sciences is multivariate. In this book, we study the asymptotic behavior of multivariate extreme of order statistics from independent and identical random vectors also the asymptotic behavior of the random maximum of independent and non-identical random vectors with random sample size. When the random sample size is assumed to be independent of the basic variables and its distribution function is assumed to converge weakly to a non-degenerate limit, the necessary and sufficient conditions for the weak convergence of the random maximum are derived.