Randomness: what is it and why does it matter?

Abstract: Randomness is a crucial resource for a broad range of important applications, such as Monte Carlo simulation and computation, generative artificial intelligence and cryptography. But what is randomness? A widely accepted definition has eluded researchers thus far, yet without one any work that relies on notion of randomness lacks scientific rigor and its results are questionable. Here, I propose an information-theory-based definition of randomness which, unlike previous attempts, does not list desired properties of the generated number sequence, but rather focuses on the physical process of random number generation itself. This approach results in a definition which complies with our intuitive perception of randomness. It is demonstrated to be non-empty and verifiable. Moreover, a new quantity named "randomness deviation" allows for a practical measure of quality of a random number generating process or a device. An expression for it is derived for a Markovian process, which is frequently encountered in practice. Finally, a process-oriented definition of a random number sequence completes the toolbox needed for understanding, proving, and practical use of randomness.