The optimal overflow and underflow probabilities with variable-length coding for the general source

In variable-length coding, the probability of codeword length per source letter being above (resp. below) a prescribed threshold is called the overflow (resp. the underflow) probability. In this study, we show that the infimum achievable threshold given the overflow probability exponent r always coincides with the infimum achievable fixed-length coding rate given the error exponent r, without any assumptions on the source. In the case of underflow probability, we also show the similar results. From these results, we can utilize various theorems and results on the fixed-length coding established by Han for the analysis of overflow and underflow probabilities