We introduce a definition of the notion of compressibility for infinite deterministic and i.i.d. random sequences which is based on the asymptotic behavior of truncated subsequences. For this purpose, we use asymptotic results regarding the distribution of order statistics for heavy-tail distributions and their link with alpha-stable laws for1 < alpha < 2. In many cases, our proposed definition of compressibility coincides with intuition. In particular, we prove that heavy-tail (polynomial decaying) distributions fulfill the requirements of compressibility. On the other hand, exponential decaying distributions like Laplace and Gaussian do not. The results are such that two compressible distributions can be compared with each other in terms of their degree of compressibility.