Large Deviations Performance of Knuth-Yao algorithm for Random Number Generation

We investigate large deviations performance of the algorithm for random number generation proposed by Knuth and Yao. First, we show that the number of input fair random bits per the length of output sequence approaches to the of output source almost surely, and the large deviations performance of this algorithm is equal to the one of the interval algorithm proposed by Han and Hoshi. Next, we consider to obtain the fixed length of output sequence from the input fair random bits with fixed length. We show that the approximation error measured by the variational distance vanishes exponentially at the speed equal to the interval algorithm as the length of output sequence tends to infinity, if the number of input fair bits per output sample is above the entropy of source. Contrarily, the approximation error measured by the variational distance approaches to two exponentially at the speed equal to the interval algorithm, if the number of input fair bits per output sample is below the entropy. ∗Dept. of Electrical and Electronic Eng., Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan