Asymptotics of stability thresholds

We study asymptotic behavior of the stability thresholds of a big line bundle, and prove explicit bounds on the error terms. This answers Jin–Rubinstein–Tian’s questions affirmatively. A key step in our proof is to show that the stability thresholds of a big line bundle can always be computed by quasi-monomial valuations. This generalizes Blum–Jonsson’s result on the stability thresholds of an ample line bundle.