 In this section we will investigate the Likelihood Ratio Convergence Theorem. This theorem shows that under certain reasonable conditions, when hypothesis \(h_i\) (in conjunction with auxiliaries in b) is true and an alternative hypothesis \(h_j\) is empirically distinct from \(h_i\) on some possible outcomes of experiments or observations described by conditions \(c_k\), then it is very likely that a long enough sequence of such experiments and observations c\(^n\) will produce a sequence of outcomes \(e^n\) that yields likelihood ratios \(P[e^n \pmid h_{j}\cdot b\cdot c^{n}] / P[e^n \pmid h_{i}\cdot b\cdot c^{n}]\) that approach 0, favoring \(h_i\) over \(h_j\), as evidence accumulates (i.e., as n increases). This theorem places an explicit lower bound on the “rate of probable convergence” of these likelihood ratios towards 0. That is, it puts a lower bound on how likely it is, if \(h_i\) is true, that a stream of outcomes will occur that yields likelihood ratio values against \(h_j\) as compared to \(h_i\) that lie within any specified small distance above 0. 