Limit joint distribution of the statistics of «Monobit test», «Frequency Test within a Block» and «Binary Matrix Rank Test»

In the case when the tested sequence consists of independent random variables having a Bernoulli distribution with the 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:math> p=1/2 the limit joint distribution of the statistics T 1 , T 2 , T 3 of the following three tests of the NIST package is obtained: «Monobit Test», «Frequency Test within a Block» and «Binary Matrix Rank Test». Necessary and sufficient conditions for asymptotic uncorrelatedness and/or asymptotic independence of these statistics are obtained. It is proved that the covariance matrix C = ‖ C ij ‖ of the limit distribution of the vector ( T 1 , T 2 , T 3 ) satisfies the relations 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 for a wide class of values <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 .