In this paper (which can be thought of as a continuation of our previous article “On several types of convergence and divergence in Archimedean Riesz spaces”) we extend the Ornstein ratio ergodic theorem to a class of Archimedean Riesz spaces. Let E be an Archimedean Riesz space, let Ẽ be the Dedekind completion of E (we may and do think of E as being a Riesz subspace of Ẽ), and let T: E → E be a positive linear operator. Let u ϵ E, u ≠ 0, let I( T, u) be the ideal in Ẽ generated by the set {T nu ¦ n ϵ N ∪ {0}} , and let I 00 ∗(T, u) be the Riesz space of all order continuous linear functionals on I( T, u). We say that u has property P if I 00 ∗(T, u) separates the points of I( T, u). Now let ƒ, g ϵ E, ƒ ⩾ 0, g ⩾ 0, and set u n = ∑ i = 0 n T iƒ, v n = ∑ i = 0 n T ig for every n ϵ N ∪ {0} . Let B be the projection band in Ẽ generated by the set {T ng ¦ n ϵ N ∪ {0}} , let B d ( B) be the band of total ratio individual divergence of the sequence (( u n , v n )) n ϵ N ∪ {0} with respect to B, and assume that g has property P . The main result of the paper is that if the sequence (( u n , v n )) n ϵ N ∪ {0} does not ratio converge individually on B, then for every u ϵ B d ( B), u ≠ 0, there exist v ϵ B( u), v ≠ 0, and a sequence of modifications ( ω n ) n ϵ N of ƒ + g such that ( ω n ) n ϵ N diverges individually to ∞ on B( v) (where B( u) and B( v) are the bands in Ẽ generated by the singletons { u} and { v}, respectively).