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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>.</m:mo> </m:math> p=1/2 . 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>≠</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>.</m:mo> </m:math> p ≠ 1/2 .