Motivated by intention to evaluate asymptotically multiple-burst-error-correcting codes on channels with memory, we will derive the following fact. Let {Zi } be a hidden Markov process, i. e. , a functional of a Markov chain with a finite state space, and Wb(Z1Z2Zn) denote the number of burst errors that appear in Z1Z2Zn, where the number of burst errors is counted using Gabidulin's burst metric , 1971. As the main result, we will prove the almost sure convergence of relative burst weight Wb(Z1Z2Zn)/n, i. e. , the relative frequency of occurrence of burst errors, for a broad class of functionals { Zi } of finite Markov chains. Functionals of Markov chains are often adopted as models of the noises on channels, especially on burst-noise channels, the most famous model of which is probably the Gilbert channel proposed in 1960. Those channel models described with Markov chains are called channels with memory (including channels with zero-memory, i. e. , memoryless ones). This work's achievement enables us to extend Gilbert's code performance evaluation in 1952, a landmark that offered the well-known Gilbert bound, discussed its relationship to the (memoryless) binary symmetric channel, and has been serving as a guide for the-Hamming-metric-based design of error-correcting codes, to the case of the-burst-metric-based codes (burst-error-correcting codes) and discrete channels with or without memory.