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We first consider a real random variable X represented through a random pair (R, T ) and a deterministic function u as X = R · u(T ). Under quite weak assumptions we prove a limit theorem for (R, T ) given X > x, as x tends to infinity. The novelty of our paper is to show that this theorem for the representation of the univariate random variable X permits us to obtain in an elegant manner conditional limit theorems for random pairs (X, Y ) = R · (u(T ), v(T )) given that X is large. Our approach allows to deduce new results as well as to recover under considerably weaker assumptions results obtained previously in the literature. Consequently, it provides a better understanding and systematization of limit statements for the conditional extreme value models.

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