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We establish the (level-1) large deviation principles for three kinds ofmeans associated with the backward continued fraction expansion. We show that:for the harmonic and geometric means, the rate functions vanish exactly at onepoint; for the arithmetic mean, it is completely degenerate, vanishing at everypoint in its effective domain. Our method of proof employs the thermodynamicformalism for finite Markov shifts, and a multifractal analysis for the R\'enyimap generating the backward continued fraction digits. We completely determinethe class of unbounded arithmetic functions for which the rate functions vanishat every point in unbounded intervals.

Large deviation principle for harmonic/geometric/arithmetic mean of digits in backward continued fraction expansion