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We consider the problem of lossless compression of individual sequences usingfinite-state (FS) machines, from the perspective of the best achievableempirical cumulant generating function (CGF) of the code length, i.e., thenormalized logarithm of the empirical average of the exponentiated code length.Since the probabilistic CGF is minimized in terms of the R\'enyi entropy of thesource, one of the motivations of this study is to derive anindividual-sequence analogue of the R\'enyi entropy, in the same way that theFS compressibility is the individual-sequence counterpart of the Shannonentropy. We consider the CGF of the code-length both from the perspective offixed-to-variable (F-V) length coding and the perspective ofvariable-to-variable (V-V) length coding, where the latter turns out to yield abetter result, that coincides with the FS compressibility. We also extend ourresults to compression with side information, available at both the encoder anddecoder. In this case, the V-V version no longer coincides with the FScompressibility, but results in a different complexity measure.

On empirical cumulant generating functions of code lengths for individual sequences