This paper investigates the problem of variable-length intrinsic randomness for a general source. For this problem, we can consider two performance criteria based on the variational distance: the maximum and average variational distances. For the problem of variable-length intrinsic randomness with the maximum variational distance, we derive a general formula of the average length of uniform random numbers. Further, we derive the upper and lower bounds of the general formula and the formula for a stationary memoryless source. For the problem of variable-length intrinsic randomness with the average variational distance, we also derive a general formula of the average length of uniform random numbers.

This paper considers a joint channel coding and random number generation from the channel output. Specifically, we want to transmit a message to a receiver reliably and at the same time the receiver extracts pure random bits independent of the channel input. We call this problem as the joint channel coding and intrinsic randomness problem. For general channels, we clarify the trade-off between the coding rate and the random bit rate extracted from the channel output by using the achievable rate region, where both the probability of decoding error and the approximation error of random bits asymptotically vanish. We also reveal the achievable rate regions for stationary memoryless channels, additive channels, symmetric channels, and mixed channels.

We consider the intrinsic randomness problem for correlated sources. Specifically, there are three correlated sources, and we want to extract two mutually independent random numbers by using two separate mappings, where each mapping converts one of the output sequences from two correlated sources into a random number. In addition, we assume that the obtained pair of random numbers is also independent of the output sequence from the third source. We first show the δ-achievable rate region where a rate pair of two mappings must satisfy in order to obtain the approximation error within δ ∈ [0,1), and the second-order achievable rate region for correlated general sources. Then, we apply our results to non-mixed and mixed independently and identically distributed (i.i.d.) correlated sources, and reveal that the second-order achievable rate region for these sources can be represented in terms of the sum of normal distributions.