We prove general invariance principles for integral functions of the empirical process. As corollaries we derive probabilistic proofs of the sufficiency criteria for a distribution to belong to the domain of attraction of the normal and stable laws with index $0 < \alpha < 2$. In the process we obtain equivalent statements of these criteria in terms of the tail behaviour of the underlying quantile function. We also give a representation of any stable random variable with index $0 < \alpha < 2$ in terms of a linear combination of two independent and identically distributed Poisson integrals. The role of a fixed number of extreme terms is exactly determined.
"Normal and Stable Convergence of Integral Functions of the Empirical Distribution Function." Ann. Probab. 14 (1) 86 - 118, January, 1986. https://doi.org/10.1214/aop/1176992618