The serial test for sampling numbers and other tests for randomness

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Abstract
In the serial test for sampling numbers (3, 4) an expression is used. It is in the form of a sum of squares and has previously been supposed to have asymptotically a χ2 (γ-variate) distribution. By evaluating it is shown that this supposition cannot be correct, since is not equal to the number of degrees of freedom. The evaluation is based on some results which can be used for testing a wide class of properties of random sequences. The definition of is generalized to deal with subsequences of length ν (instead of length 2) and the generalized , or is then decomposed into a linear form of simple variates defined in terms of discrete Fourier transforms. When the number, t, of digits in the sample space is prime these simple variates are shown to have asymptotically independent χ2 distributions with one degree of freedom each. The coefficients of the linear form are not 1 but they become 1 when the first and second differences with respect to ν are taken. Thus and have asymptotically χ2 distributions. The work is closely related to that of Bartlett (1) and Vajda (6).

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