Limit joint distribution of the statistics of the “Monobit Test”, “Frequency Test within a Block” tests and the generalized “Serial Test” of the NIST package

Abstract We find the limit joint distribution of the statistics T 1, T 2, T 3 of the “Monobit Test”, “Frequency Test within a Block”, and the generalized “Serial Test” tests of the NIST package in the case where the sequence under consideration consists of independent Bernoulli random variables with parameter p = 1 2. p=12. We show that T 1 and (T 2, T 3) are asymptotically uncorrelated, T 2 and T 3 are asymptotically positively correlated, and T 1, T 2, T 3 are pairwise asymptotically dependent. We also prove that the covariance matrix C of the limit distribution of the vector (T 1, T 2, T 3) satisfies C 12 = C 21 = C 13 = C 31 = 0, C 23 = C 32 > 0. The limit behavior of the vector (T 1, T 2, T 3) is described in the case p ≠ 1 2. p 12.