A note on convergence rate of constrained capacity estimation algorithms over ISI channels

It has recently become popular to use simulation-based algorithms to empirically estimate achievable information rates over intersymbol interference (ISI) channels with inputs from specific input constellations. Such algorithms are guaranteed to converge by invoking the Shannon-McMillan-Brieman theorem provided that the output sequence is stationary and ergodic. In this note, we establish a central limit theorem result on the rate of convergence, and show that the variance of the estimates decreases like 1/N (where N is the sequence length employed) as N goes to infinity. This result indicates that it is possible to achieve estimation accuracy with any desired level by simply increasing the number of samples appropriately.