The first part of this talk surveys different notions of randomness of one-dimensional sequences. A focal point will be on what is not known about local statistics which go beyond uniform distribution modulo one. The second part is reporting on joint work with Christopher Lutsko and Athanasios Sourmelidis. By employing Fourier analytic tools, this work provides a better understanding of how random slowly growing monomial sequences are - showing that their correlations are Poissonian.