Weak variable-length source coding theorems

The author first defines a general source as an infinite sequence X={Xⁿ=(X₁⁽ⁿ⁾,...,X_n⁽ⁿ⁾ )}_n=1^∞ of n-dimensional random variables Xⁿ where each component random variable X_i ⁿ (1⩽i⩽n) takes values in a countably infinite set 𝒳 that is called the source alphabet. It should be noted that each component of Xⁿ may change depending on block length n. This implies that the sequence X is quite general in the sense that it may not satisfy even the consistency condition as usual processes, where the consistency condition means that for any integers m,n such that m<n it holds that X_i^(m)≡X_i⁽ⁿ⁾ for all i=1,2,...,m. The class of sources thus defined covers a very wide range of sources including all nonstationary and/or nonergodic sources