Lemma 34 (corrected: the pathwise bound $|\hat S^t_n-s(\beta)|\le\sum_{i=0}^{t-1}\kappa^iD_{t-i}+\kappa^tC$)

Let $(\hat S^t_n)_{t\ge0}$ be the process (4.1) from an initial threshold $\hat S^0_n\in\mathbb R$, let $\kappa\ge0$ be a Lipschitz constant of $q(P(\beta,\cdot))$, let $s(\beta)$ be a fixed point of $q(P(\beta,\cdot))$, and let

Then on every outcome and for every $t\ge0$,

The printed Lemma 34 states that $\sum_{i=1}^t z^{t-i}\kappa^i+\kappa^tC$ stochastically dominates $|\hat S^t_n-s(\beta)|$. That is false (see below). Its proof, (I.18)–(I.24) on p.54, derives the recursion $|\hat S^t_n-s(\beta)|\le D_t+\kappa|\hat S^{t-1}_n-s(\beta)|$; unrolled, it gives this pathwise bound, which carries weight $\kappa^0$ on the newest error. Together with Lemma 33 (corrected) and a union bound it proves Theorem 4.

Differences from the source:

• correction: $\sum_{i=1}^{t}z^{t-i}\kappa^i+\kappa^tC$ stochastically dominates $|\hat S^t_n-s(\beta)|$ → $|\hat S^t_n-s(\beta)|\le\sum_{i=0}^{t-1}\kappa^iD_{t-i}+\kappa^tC$ on every outcome; reason: the printed lemma is false: it inherits the failure of Lemma 33, and its sum is mis-indexed. In the instance of Lemma 33 (corrected) with $t=1$ and $\hat S^0_n=s(\beta)$, $C=0$ and the printed bound is $\kappa z^0$ with $\kappa\le0.0025$, while $|\hat S^1_n-s(\beta)|\ge3$ with probability about $6.7\cdot10^{-11}$. The corrected statement is the recursion (I.18)–(I.24) unrolled, with the sum from $i=0$
• generalization: the conditions of Theorem 4; $\kappa$ the Lipschitz constant of $q(P(\beta,s))$ → no model hypotheses; any Lipschitz constant $\kappa\ge0$ and any fixed point; reason: the pathwise inequality uses only the Lipschitz property and the fixed-point equation; the source's setting is a special case
• generalization: Assumption 2: the $(Z,c)$-marginal of $F$ has finite support; $c_i\in\mathcal C\subset L^2(\tilde{\mathcal X},dx)$ with $\mathcal C$ convex and compact; outcomes $Y_i(0),Y_i(1)$ (pp.6, 8) → finitely many types $i\in\iota$ with weights $w_i>0$, $\sum_i w_i=1$; no membership in $\mathcal C$ is required and the outcomes are not modelled; reason: no statement of the mission mentions $\mathcal C$ or the outcomes, and the score law does not depend on them; every instance of the source is an instance of the Lean model
• reading: the $q$-th quantile $q(\cdot)$ → the lower quantile $\inf\{r : q\le G(r)\}$; reason: the convention of Shao (2003), cited for Theorem 17; for $P(\beta,s)$ it is the unique inverse of Lemma 30
• reading: the process (4.1); the running text of p.9 prints $\hat S^t_n=q(P^{n,t-1})$ → display (4.1): $\hat S^t_n=q(P^n(\beta,\hat S^{t-1}_n))$, where cohort $t$ is a fresh sample, independent over all $(t,j)$, each agent with law $F$, responding to the realized $\hat S^{t-1}_n$; reason: display (4.1) is the definition and the text says that $n$ new agents are sampled i.i.d. at each step