Mismatched asymptotic equipartition property ($n^{-1}{\log }_2{q}_n(X^n)\to -H(p,q)$)

For independent and identically distributed random variables with probability mass function $p$ on a finite alphabet, evaluate their realized strings under a fixed product model $q_n(x^n)={\prod }_{i}q(x_i)$. If $q$ is positive on the probability support of $p$, the strong law of large numbers givesalmost surely, using cross-entropy equals entropy plus relative entropy. If $q$ vanishes at an outcome of positive $p$ probability, that outcome eventually appears almost surely, making all subsequent product likelihoods zero and the limit $-\infty$. This matches infinite cross-entropy.