Probability Theory & Stochastic Processes Question

Prove that the randomness (or where / how it starts  to become random) of a one-dimensional array in variable or infinite  length such that whenever the real number detected or received, develop and apply a simple rule to validate your decision whether the data is random or not; in real time (from the beginning to the last one, just arrived) with given length (without using histograms or randomness tests) and plot the necessary graphics. Proofing Process should be starting with the extensive probability theory (showing relevant theorems with their proofs and necessary calculations to be able to apply them to the problem). 

Codes (python or Matlab) would be good to be provided to plot the requested graphic(s), if applicable.

Don't use ChatGPT!