LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Integrated and conditional uniform convergence of Cox risk-set sums

Statement

Let $O_{i} = (T_{i}, C_{i}, Z_{i} (\cdot))$ , $i \geq 1$ , be IID subject records with law $P$ , with jointly measurable predictable covariates, $p < \infty$ , and $∥ Z_{i} (t) ∥_{2} \leq K$ for all $i, t$ outside one null event. Let $B \subset R^{p}$ be compact and contain $β_{0}$ . With $Y_{i} (t) = 1 {U_{i} \geq t}$ , define, for $r = 0, 1, 2$ , $S_{n}^{(r)} (t, β) = \frac{1}{n} \sum_{i = 1}^{n} Y_{i} (t) Z_{i} (t)^{\otimes r} e^{β^{T} Z_{i} (t)}, s^{(r)} (t, β) = E_{P} [Y_{i} (t) Z_{i} (t)^{\otimes r} e^{β^{T} Z_{i} (t)}],$ where $Z^{\otimes 0} = 1$ , $Z^{\otimes 1} = Z$ , and $Z^{\otimes 2} = Z Z^{T}$ . Suppose $c_{0} : = \inf_{t \leq τ} s^{(0)} (t, β_{0}) > 0$ . For every finite deterministic measure $ν (d t) = w (t) d t$ with nonnegative integrable density $w$ , put $D_{n} (t) = \max_{r = 0, 1, 2} \sup_{β \in B} ∥ S_{n}^{(r)} (t, β) - s^{(r)} (t, β) ∥_{\max} .$ Then for each fixed $t$ , $E [D_{n} (t)^{2}] \to 0$ , and $E \int_{0}^{τ} D_{n} (t)^{2} ν (d t) ⟶ 0.$ There are measurable events $E_{n}$ with $P (E_{n}) \to 1$ and a constant $c > 0$ such that on $E_{n}$ , $S_{n}^{(0)} (t, β) \geq c for every t \leq τ, β \in B .$ Define ${\overset{ˉ}{Z}}_{n} = S_{n}^{(1)} / S_{n}^{(0)}$ and $V_{n} = S_{n}^{(2)} / S_{n}^{(0)} - {\overset{ˉ}{Z}}_{n} {\overset{ˉ}{Z}}_{n}^{T}$ , setting each to zero at a zero denominator, and define $\overset{ˉ}{z} = s^{(1)} / s^{(0)}$ , $V = s^{(2)} / s^{(0)} - \overset{ˉ}{z} {\overset{ˉ}{z}}^{T}$ . Their fixed-time and integrated parameter-uniform errors satisfy $E \sup_{β \in B} ∥ {\overset{ˉ}{Z}}_{n} (t, β) - \overset{ˉ}{z} (t, β) ∥_{2}^{2} \to 0, E \sup_{β \in B} ∥ V_{n} (t, β) - V (t, β) ∥_{\max}^{} \to 0,$ $E \int_{0}^{τ} \sup_{β \in B} ⁣ {∥ {\overset{ˉ}{Z}}_{n} - \overset{ˉ}{z} ∥_{2}^{2} + ∥ V_{n} - V ∥_{\max}^{}} ν (d t) ⟶ 0.$ These conclusions require no regularity of the time-indexed covariate class beyond bounded predictability. In particular take $ν (d t) = λ_{0} (t) d t$ .

The stronger uniform-in-time conclusion remains valid under the following additional, sufficient condition: every scalar coordinate class $F_{r, k} = {O \mapsto [Y (t) Z (t)^{\otimes r} e^{β^{T} Z (t)}]_{k} : t \leq τ, β \in B}$ is pointwise measurable and has finite $L^{1} (P)$ bracketing number at every positive width. Under that condition, $\sup_{t} D_{n} (t) \to_{P} 0$ , and both ratio errors converge uniformly on $[0, τ] \times B$ in probability. This optional condition is not imposed by the Cox inference consumers.

Facts & Assumptions

Given: The IID bounded predictable subject records, compact $B$ , positive population risk support, and finite deterministic measure $ν$ of the Statement. Only the final conditional conclusion assumes pointwise measurability and finite bracketing.

[F1]

Cox proportional-hazards model and partial likelihood: $U_{i} = \min (T_{i}, C_{i})$ and $Y_{i} (t) = 1 {U_{i} \geq t}$ .

[F2]

Empirical, centered, and Rademacher processes: $P_{n} f = n^{- 1} \sum_{i} f (O_{i})$ and $P f = E_{P} f$ .

[F3]

Covering, packing, bracketing numbers, and metric entropy: finite $L^{1} (P)$ brackets have measurable endpoints and the specified positive widths.

[F4]

Glivenko–Cantelli classes: pointwise-measurable classes have measurable suprema and their GC law is convergence in probability.

[F5]

Glivenko–Cantelli theorem by bracketing entropy: an IID pointwise-measurable class with integrable envelope and finite $L^{1} (P)$ bracketing numbers is GC in probability.

[F6]

Progressively measurable and predictable processes: predictable processes are jointly measurable, and deterministic Borel time functions are predictable.

[F7]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative product-measurable functions can be integrated in either order.

[F8]

Dominated convergence: pointwise convergence under an integrable majorant gives convergence of integrals.

[A1]

Only for the final conditional conclusion, the classes $F_{r, k}$ have pointwise-measurable versions and finite bracketing numbers at every positive width.

[A2]

AC is inherited through the Cox predictability interface; finite nets and probability bounds below add no further choice assumption. The Axiom of Choice

Proof

technique · Use bounded IID coordinate variances and finite parameter nets at each time, integrate their errors, and protect denominators by the monotone unweighted risk count. Retain the bracketing law as an optional stronger conclusion

1.1givenA2F1F6F7algebra

Put $R = \sup_{β \in B} ∥ β ∥_{2}$ and $C = e^{K R} \max (1, K, K^{2})$ . Each risk coordinate is bounded by $C$ and is Lipschitz in $β$ with constant at most $K C$ , since the segment between two points of $B$ lies in the radius- $R$ ball. By [F6]–[F7], the expectations are jointly measurable in time and parameter. Both empirical and population functions are continuous in parameter, so each parameter supremum equals the supremum over a fixed countable dense subset of $B$ and is measurable. Population ratios are deterministic Borel time functions and hence predictable.

2.1F2F7F8step 1.1constructalgebra

For each fixed $t, β$ and coordinate $f$ , independence and centering give $E [(P_{n} f - P f)^{2}] = Var (f) / n \leq C^{2} / n$ : the cross terms vanish by independence. On a finite $ε$ -net of $B$ , the squared maximum over its $m_{ε}$ points and the $d = 1 + p + p^{2}$ coordinates has expectation at most $d m_{ε} C^{2} / n$ . Between a point and its nearest net point, the empirical-minus-population Lipschitz bound is $2 K C ε$ . Thus ${lim sup}_{n} E D_{n} (t)^{2} \leq 8 K^{2} C^{2} ε^{2}$ ; let $ε ↓ 0$ . The uniform bound $D_{n} \leq 2 C$ and [F8] applied to $t \mapsto E D_{n} (t)^{2}$ , followed by [F7], prove $E \int D_{n}^{2} d ν \to 0$ .

2.2givenF1step 1.1algebra

Write $q = P (U_{i} \geq τ)$ . Since $c_{0} \leq s^{(0)} (τ, β_{0}) \leq e^{K ∥ β_{0} ∥_{2}} q$ , $q > 0$ . For $Q_{n} = n^{- 1} \sum_{i} 1 {U_{i} \geq τ}$ , independence gives $E (Q_{n} - q)^{2} = q (1 - q) / n$ , so $P (Q_{n} < q / 2) \leq 4 (1 - q) / (n q) \to 0$ . Monotonicity of $Y_{i}$ and the weight bound give $S_{n}^{(0)} (t, β) \geq e^{- K R} Q_{n}$ for every $t, β$ . Hence on the event $E_{n} = {Q_{n} \geq q / 2}$ all sample denominators are at least $c = e^{- K R} q / 2$ , while all population denominators are at least $e^{- K R} q$ .

3.1F7F8step 1.1step 2.1step 2.2algebra

On $E_{n}$ , ratio subtraction and the bounds of steps 1.1 and 2.2 bound both $\sup_{B} ∥ {\overset{ˉ}{Z}}_{n} - \overset{ˉ}{z} ∥_{2}$ and $\sup_{B} ∥ V_{n} - V ∥_{\max}$ by a constant times $D_{n} (t)$ . Off $E_{n}$ , the weighted means have norm at most $K$ and the second-moment matrices have entries at most $K^{2}$ , including the zero convention, so both errors remain bounded by constants depending only on $K, p$ . Their expected squared errors are therefore at most $C_{1} E D_{n} (t)^{2} + C_{2} P (E_{n}^{c})$ . Steps 2.1–3.1 prove the fixed-time conclusions; integration against finite $ν$ , or [F7]–[F8], proves their integrated versions.

4.1A1F3F4F5step 1.1step 2.2step 3.1∎

Under [A1], each coordinate class has the envelope from step 1.1 and the hypotheses of [F3]–[F5]. Apply only the probability branch of [F5] and take the finite union over coordinates to obtain $\sup_{t} D_{n} (t) \to_{P} 0$ . Step 3.1's ratio bound on $E_{n}$ , whose probability tends to one, gives both uniform-in-time ratio conclusions. No such time-indexed class hypothesis was used in steps 1.1–4.1.

Remarks

Bounded predictability alone does not imply the optional uniform-in-time law. The controlling-plan reconciliation and an explicit IID predictable counterexample with positive information are recorded in research/statistical-models-experiments-62-owner-cox-uniform-resolution.json. The integrated law above preserves arbitrary bounded predictable covariates in the estimator expansion.

Depends on

Used by

Dependency tree · two levels

43 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lu Tian, STAT331 Revisiting Cox's PH Model: Asymptotic Theory, PDF pp. 1–5 and 9–10 (standard reference, not scraped)
Qiqing Yu, Survival Analysis (Lecture Notes), §5.2, PDF pp. 58–63 (standard reference, not scraped)