Almost Sure Convergence of Relative Frequency of Occurrence of Burst Errors on Channels with Memory∗

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(Z1Z2 . . . Zn) denote the number of burst errors that appear in Z1Z2 . . . Zn, where the number of burst errors is counted using Gabidulin’s burst metric [1], 1971. As the main result, we will prove the almost sure convergence of relative burst weight Wb(Z1Z2 . . . Zn)/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 [2] 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 [3] 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-errorcorrecting codes) and discrete channels with or without memory. key words: burst-error-correcting code, burst metric (weight), Markov chain, law of large numbers, swept covering