Variable-Length Intrinsic Randomness Allowing Positive Value of the Average Variational Distance

This paper considers the problem of variable-length intrinsic randomness. The performance criteria are the average variational distance and the mean length. This problem has a dual relationship with variable-length resolvability. Previous study has derived the general formula of the $\epsilon$-variable-length resolvability. We derive the general formula of the $\epsilon$-variable-length intrinsic randomness. Namely, we characterize the supremum of the mean length under the constraint the value of the average variational distance is smaller than or equal to some constant. Our result clarifies the dual relationship between the general formula of $\epsilon$-variable-length resolvability and that of $\epsilon$-variable-length intrinsic randomness. We also derive the lower bound of the quantity characterizing our general formula.