6.6 Aldous’s criterion
\(\operatorname {aldousQ}(P,X,\mathcal{F},d,e)\) is the supremum, over real-valued \(\mathcal{F}\)-stopping times \(\tau \le 1\) and deterministic shifts \(0\le \delta \le d\), of \(P\{ \, e\le |X_{\min (\tau +\delta ,1)}-X_\tau |\, \} \), valued in \([0,\infty ]\). Note the truncation: the shifted time is \(\min (\tau +\delta ,1)\), so the increment never looks past the horizon (for Skoro-valued processes this is consistent with the flat-right encoding). The quantity is monotone in the shift budget \(d\), and any single admissible pair \((\tau ,\delta )\) bounds it from below (le_aldousQ_of_stoppingTime).
Let \((X_i)_{i\in \iota }\) be \(D\)-valued measurable random elements on probability spaces \((\Omega _i,P_i)\), each adapted to a right-continuous filtration \(\mathcal{F}_i\). Assume (i) uniform sup-norm tightness: for every \(\eta {\gt}0\) there is \(a\) with \(P_i\{ a\le \operatorname {supNorm}X_i\} \le \eta \) for all \(i\); and (ii) the Aldous condition: for all \(\varepsilon ,\eta {\gt}0\) there is \(\delta {\gt}0\) with \(\alpha _i(\delta ,\varepsilon )\le \eta \) uniformly in \(i\). Then the family of laws \(\{ (P_i)_*X_i\} \) is tight on \(D\). The proof is Aldous’s genuine two-scale argument (Billingsley (16.24)ff): the measurable witness superset for the modulus level sets is an interior bad set of close consecutive crossings plus a boundary shift-average detector, whose masses are controlled by \(\alpha \) at two scales via a raw interval Lebesgue average over the shift; a single-scale variant was refuted by counterexample during formalization.
Second-moment form of the criterion: if, uniformly over the family, over stopping times \(\tau \le 1\) and shifts \(0\le \delta \le d\), the truncated increments satisfy \(\mathbb {E}\bigl[(X_{\min (\tau +\delta ,1)}-X_\tau )^2\bigr]\le M(d)\) with \(M(d)\to 0\) as \(d\to 0^+\) (plus a.e.-measurability of the increments and the uniform sup-norm bound), then the laws are tight on \(D\). It combines Chebyshev’s inequality inside the Aldous supremum (aldousQ_le_of_second_moment) with Theorem 6.15.