Accumulating ratings → positive sampling rates
g(theta)>0. This makes explicit the
growing-sample regime used by the paper’s convergence and
large-deviation arguments
(cited publication:669–670, 745).
Independent rating histories → a product probability
law. This realizes the independence used in the Appendix’s
pairwise factorization; the ranking objective uses only those pairwise
marginals (cited publication:1638–1642).
Theorem 1 reading
The printed adjacent-pair minimum through i=M-1 →
0<=i<=M-2, where M is the number of
ordered seller types: the last printed term refers to an undefined next
type.
Strict cross-quality upper-tail comparisons at every rating → only
above the lowest rating. At the lowest rating the upper-tail probability
is one for every type, so it cannot be strictly increasing.
Real-valued intermediate rate costs → extended-real Legendre costs
inside the infimum, so thresholds outside rating support have infinite
cost. The final minimum is finite. This avoids a terminal full-support
assumption and retains the stated exponential rate for the ranking
error.
Displayed indices and
finite-state route
In the aggregate-score display, n_k(theta)+1 summands
indexed through n_k(theta) → exactly
n_k(theta) ratings, indexed from zero through
n_k(theta)-1.
The Appendix’s continuum substitution → a finite iid probability
comparison with a fixed-constant event sandwich before taking normalized
logarithms. This supplies the discrete probability rate without treating
the continuum integral as an exact finite probability.
Population-state boundary
Theorem 1’s population recurrence → an explicitly declared iid
rating law. The formalization has not connected that law to the printed
recurrence, whose transition uses n_k and
n_(k-1) while updating mu_k to
mu_(k+1). This is an unproved model connection, not a
counterexample to the ranking-rate theorem.