Large Deviations Performance of Interval Algorithm for Random Number Generation

We investigate large deviations performance of interval algorithm for random number generation. First, we show that the length of input sequence per the length of output sequence approaches to the ratio of entropies of input and output distributions almost surely. Next, we investigate large deviations performance especially for intrinsic randomness. We show that the length of output fair random bits per input sample approaches to the entropy of the input source almost surely, and we can determine the exponent in this case. Further, we consider to obtain the fixed number of fair random bits from the input sequence with fixed length. We show that the approximation error measured by the variational distance and divergence vanishes exponentially as the length of input sequence tends to infinity, if the number of output random bits per input sample is below the entropy of the source. Contrarily, the approximation error measured by the variational distance approaches to two exponentially and the approximation error measured by the divergence approaches to infinity linearly, if the number of random bits per input sample is above the entropy of the source. ∗Dept. of Electrical and Electronic Eng., Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan