Eigenvalue of pseudorandomnumber generator determining randomness of random sequence

A theoretical analysis of a recently proposed randomness test is given which is based on the ensemble average technique. In this technique, the ergodicity of the transformation to form a random sequence is assumed so that the Perron-Frobenius integral operator is fully utilized. The Galerkin approximation to the integral operator is also introduced which provides a finite dimensional matrix. Numerical experiments of three tests for several transformations show that the magnitude of the second largest eigenvalue of the matrix plays an important role in determining randomness of the random sequence