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We consider a real random variable X represented through a random pair ofreal random variables (R,T) and a deterministic function u as X=Ru(T). Undersome additional assumptions, we prove a limit theorem for (R,T) given X>x, as xtends to infinity. As a consequence, we derive conditional limit theorems forrandom pairs (X,Y)=(Ru(T),Rv(T)) given that X is large. These results implyearlier ones which were obtained in the literature under stronger assumptions.

A conditional limit theorem for a bivariate representation of a univariate random variable and conditional extreme values