{"id":"03c9b69d-470f-47ac-9a5e-84596404c1e5","arxiv_id":"2506.21172","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For dependent, approximately stationary random functions in C0, the partial sum process is within O(N^{-τ}) of a functional Brownian motion in Prokhorov and Wasserstein distance.","lead":"This paper proves new finite-sample bounds on the distance between the partial sum process of sparsely observed functional data and its Brownian limit, with polynomial rates. The result yields couplings and validates open-ended monitoring schemes for functional time series.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Step 2 of Theorem 4.1 rests on an unverified external Gaussian approximation (Lemma B.2 / Theorem 3.27 [40]) whose hypotheses are not checked for the nonstationary triangular array with growing dimension.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: Lemma A.3's Gaussian approximation step relies on an external theorem whose applicability to the paper's nonstationary, growing-dimensional triangular array is not documented. I agree that this is the pivotal point for Theorem 4.1. I also reviewed the other issues noted by the reader: the false pointwise inequality in the proof of Corollary 2 and the notation inconsistency in Theorem 6.1 are real but affect corollaries and the monitoring application rather than the main theorem. The central claim can stand only if Lemma B.2 is a correct consequence of Theorem 3.27 in [40] and actually applies to the grid discretization array. Since the paper does not provide the cited theorem's statement or a proof of Lemma B.2, the result is plausible but not fully verifiable as written. This supports the conditional verdict: the theorem should be accepted only after the external Gaussian approximation is reproduced and its hypotheses are checked. No change to the reader's CONDITIONAL verdict is needed.","tokens_in":41393,"tokens_out":13355,"duration_ms":158122,"concrete_test":"Obtain the exact statement of Theorem 3.27 from Dehling, Mikosch and Sørensen (2002) and independently re-derive Lemma B.2 from it. Then specialize the lemma to the triangular array in Lemma A.3: set q = |G_N| ~ N^{rho+2sigma} and v_i^N[d(x)] = X_{i,M_N}(u) * 1{i <= floor(lambda N)}. Check (a) whether the theorem permits nonstationary triangular arrays with dimension q -> inf, or whether the array can be legitimately blocked or reshaped; (b) whether the theorem gives a bound for the Prokhorov metric with respect to the max norm, not merely another distance; (c) whether the error N^{-1/20} q^3 is valid under alpha-mixing with eta2 > 3 + 9/eta1 and the indicated dependence on q.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 4.1, and its proof hinges on Step 2 (Lemma A.3), which reduces the discretized process to a polynomial Prokhorov bound by invoking Lemma B.2. Lemma B.2 is asserted to be 'a direct consequence of Theorem 3.27 in [40]', but the paper neither reproduces that theorem nor verifies its hypotheses against the actual array. Lemma B.2 is stated for a fixed sequence v1, v2, ... in R^q with fixed q, while Lemma A.3 applies it to a triangular array v_i^N of dimension |G_N| ~ N^{rho+2sigma}, with entries X_{i,M_N}(u) * 1{i <= floor(lambda N)}. Both the dimension and the truncation indicators depend on N, and the sequence is nonstationary. It is not shown that Theorem 3.27 of [40] covers such triangular arrays, that its moment and mixing conditions match (eta1 = J-3, eta2 = nu), or that its error has the stated form N^{-1/20} q^3 for the Prokhorov metric induced by the max norm. If the quoted theorem in fact requires stationarity, fixed dimension, stronger mixing decay, or yields a different dependence on q and N, then inequality (A.16) does not follow and the polynomial rate of Theorem 4.1 is not established. This is the most load-bearing point: every subsequent coupling result (Corollary 1, Theorem 5.1, Theorem 6.1) inherits Theorem 4.1's rate, so an unchecked gap here would invalidate the paper's main contribution.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops finite-sample bounds on the Prokhorov and Wasserstein distances between the partial sum process of reconstructed functional data and a Brownian limit in spaces of continuous functions. The central object is a partially observed, weakly dependent triangular array of sparse estimators $X_{i,M_N}$; under two assumptions (polynomial $\\alpha$-mixing, moments and H\\\"older smoothness; and fixed, expanding, or infinite domain conditions), Theorem 4.1 asserts a polynomial rate $\\pi_\\infty(P_N,W)\\le C N^{-\\tau}$. The proof proceeds in three steps: (1) approximate $P_N$ by a grid discretization using empirical-process entropy bounds; (2) approximate the discretized process by a high-dimensional Gaussian using an external Gaussian-approximation theorem; (3) approximate the discretized Brownian motion by the continuous Brownian motion using modulus-of-continuity and tail estimates. From Theorem 4.1 the paper derives couplings, Wasserstein bounds, a bounded law of the iterated logarithm, an almost sure invariance principle, and a monitoring scheme for sparse functional data.","tokens_in":41664,"tokens_out":16534,"duration_ms":184258,"significance":"If Theorem 4.1 is valid, this is a substantial contribution: it would give the first polynomial-rate Prokhorov bound for a function-valued partial sum process under polynomially decaying $\\alpha$-mixing and only approximate stationarity, and it would provide a new route to couplings and to open-ended monitoring for sparse functional data. The two-step discretization strategy is genuinely novel, and the paper is honest about the role of external benchmarks (entropy bounds, Gaussian approximation, matrix perturbation inequalities). The main strength of the presentation is the explicit separation of the three approximation steps and the detailed supplementary material. However, the central rate in Step 2 rests on an external theorem whose hypotheses are not checked, so the main claim is currently conditional on an unverified ingredient.","major_comments":[{"comment":"The proof of Lemma A.3 invokes Lemma B.2 for the bound $\\pi_\\infty(\\vec P_N^{\\mathrm{dis}}, \\mathcal N(0,\\Sigma_N)) \\le C N^{-1/20}|\\mathcal G_N|^3$ in (A.16). Lemma B.2 is stated for a fixed sequence $v_1,v_2,\\dots\\in\\mathbb R^q$ with fixed dimension $q$, whereas Lemma A.3 applies it to the triangular array $\\vec v_i^N$ whose entries contain the $N$-dependent truncation indicators $1\\{i\\le \\lfloor\\lambda N\\rfloor\\}$ and whose dimension $|\\mathcal G_N|\\asymp N^{\\rho+2\\varsigma}$ grows with $N$. The manuscript neither reproduces Theorem 3.27 of [40] nor verifies that it covers nonstationary triangular arrays with growing dimension, that its mixing and moment hypotheses match the choices $\\eta_1=J-3$, $\\eta_2=\\nu$, or that its error is of the stated form for the maximum-norm Prokhorov metric. Since (A.16) is the only source of the polynomial rate in Step 2, this is load-bearing for Theorem 4.1 and hence for Corollary 1, Theorem 5.1, Corollary 3, and Theorem 6.1. The authors need to reproduce the external theorem or replace it with a Gaussian approximation result whose hypotheses are verified directly for this array.","section":"Supplementary Information, Lemma B.2 and Lemma A.3 (Step 2)"},{"comment":"The proof says that on the coupling space one can define $X_{n,M_N}(u)=P_N^{\\mathrm{lin}}(n/N,u)-P_N^{\\mathrm{lin}}((n-1)/N,u)$. But the process $P_N^{\\mathrm{lin}}$ is normalized by $1/\\sqrt N$ in (7) and (A.7), so this difference equals $X_{n,M_N}/\\sqrt N$, not $X_{n,M_N}$. A correct construction would define $X_{n,M_N}$ as $\\sqrt N$ times the increment of $P_N^{\\mathrm{lin}}$ and then state the resulting coupling in terms of $\\sup_k \\|\\sum_{i\\le k}X_{i,M_N}-\\sqrt N\\,W(k/N)\\|$. As written, the proof of Corollary 1 and the subsequent results that rely on it (Theorem 5.1 and Remark 2) are incomplete.","section":"Section 5, Corollary 1, proof after (12)"},{"comment":"The proof applies the coupling of Theorem 4.1 to the process $P_{k_N,N}$ on $\\lambda\\in[0,1]$, but equation (25) evaluates $P_{k_N,N}((N+k)/k_N)$ for $k$ up to $k_N$; for $k=k_N$ the argument equals $1+N/k_N>1$, so the process is used outside its domain and outside the range of the approximation. There is also a scaling inconsistency: by the definition $P_{k_N,N}(\\lambda)=N^{-1/2}\\sum_{i\\le\\lambda k_N}\\varepsilon_i$, the quantity $P_{k_N,N}(N/k_N)$ equals $N^{-1/2}\\sum_{i\\le N}\\varepsilon_i=O_P(1)$, so the displayed identity $\\bar\\gamma_2=\\sqrt{k_N/N}\\,\\|P_{k_N,N}(N/k_N)\\|$ would be of order $N^{\\zeta/2}O_P(1)$, whereas the statistic $\\bar\\gamma_2=\\|N^{-1/2}\\sum_{i\\le N}\\varepsilon_i\\|$ is $O_P(1)$. The factor $\\sqrt{k_N/N}$ in (25) and in the definition of $\\bar\\gamma_2$ appears to be spurious, and the argument would need an explicit extension of the coupling to the growing interval $[0,1+N^{-\\zeta}]$ or a different decomposition. Without this, the level and consistency claims of the monitoring scheme are not established.","section":"Section 6, proof of Theorem 6.1, equations (25)–(26)"}],"minor_comments":[{"comment":"The sentence containing \"in all of of the Supplementary Information\" has a duplicated word.","section":"Supplementary Information, first paragraph of Appendix A"},{"comment":"Lemma B.1 states the bound with $\\bar\\tau=10\\rho$, but its proof derives the sharper exponent $5\\rho/2$; the text should reconcile these exponents, since Theorem 5.1 compares $\\tau'$ with $(4-q)\\tau/4$.","section":"Supplementary Information, Lemma B.1"},{"comment":"Assumption 5 is numbered (i), (iii), (iv) without a (ii); renumber for consistency.","section":"Section 6, Assumption 5"}],"recommendation":"major_revision","confidential_remarks":"The referee report and the stress-test note agree on the main risk: the Step 2 Gaussian approximation is not verified against the actual triangular array. In addition, the proof of Theorem 6.1 appears to contain a concrete scaling error in (25)–(26), and Corollary 1's proof has a missing $\\sqrt N$ factor. These are all fixable within the manuscript's scope, so I do not recommend rejection, but the central claim cannot be accepted until the external theorem is stated and its hypotheses are checked, and the monitoring proof is corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plain take: this is a serious paper with a genuinely new two-step proof strategy, and the main theorem is plausible; but the most load-bearing step in the Supplementary Information leans on an external high-dimensional Gaussian approximation whose hypotheses are never checked against the triangular array. That is the thing to verify before anyone relies on Theorem 4.1.\n\nWhat is new: the finite-sample Prokhorov/Wasserstein bounds for function-valued partial sum processes under alpha-mixing, nonstationary triangular arrays, and growing or unbounded domains. That combination is new. The discretization-then-Gaussian-approximation proof idea is also new and is a nice way around the absence of KMT approximations in infinite dimension. Section 6's open-ended monitoring application is a legitimate payoff, and the comparison table with the prior literature is honest. The self-citations in the monitoring section are to applications of the same framework, which is fine.\n\nThe soft spot is real. Lemma B.2 is stated for a fixed dimension q and a fixed sequence v1, v2, ...; the proof of Lemma A.3 applies it to a triangular array whose dimension |G_N| grows like N^{rho+2sigma} and whose entries contain the indicator 1{i <= floor(lambda N)}, so the sequence changes with N. The paper does not reproduce Theorem 3.27 of [40] and does not verify that it covers such nonstationary, growing-dimension arrays, nor that the moment and mixing conditions match exactly. If that theorem is only for stationary sequences or fixed q, then inequality (A.16) does not follow and Theorem 4.1 loses its rate. This is not cosmetic: Corollary 1, Theorem 5.1, and Theorem 6.1 all inherit that rate. The stress-test concern lands.\n\nMinor issues: Corollary 2's proof contains a pointwise inequality that is false as written, and the proof of Theorem 6.1 mixes P_{k_N,N} and P_{k_N,k_N}. Both look fixable and neither affects Theorem 4.1.\n\nBottom line: this deserves a serious referee. The central idea is worth engaging, but the paper should not be accepted until Lemma B.2's application is fully justified—either by reproducing the external theorem and checking its hypotheses, or by replacing it with a self-contained Gaussian approximation. I would not cite it yet.","headline":"A serious, genuinely new rate result whose central proof has a load-bearing gap: Lemma B.2's external Gaussian approximation is applied to a triangular array it may not cover.","tokens_in":42247,"tokens_out":2745,"would_cite":false,"duration_ms":30638,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62M10","62R10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under polynomial α-mixing and approximate stationarity, the partial sum process of reconstructed functional data is within C N^{-τ} of Brownian motion in Prokhorov distance.","keywords":["functional data analysis","sparse functional data","Prokhorov metric","Wasserstein metric","invariance principle","alpha-mixing","coupling","change point monitoring"],"falsifier":"Check the statement of Theorem 3.27 in [40] against the vectors $\\vec{v}_i$ defined in Lemma A.3: if it requires stationarity, fixed dimension, or stronger mixing than the paper's polynomial $\\alpha$-decay, then Lemma B.2 fails and Theorem 4.1 has no proof. Alternatively, simulate a simple stationary autoregressive functional process and estimate $\\pi_\\infty(P_N, W)$ for increasing $N$; an empirical decay slower than any polynomial rate would contradict the theorem.","tokens_in":41145,"feed_emoji":"📈","tokens_out":7741,"duration_ms":74116,"temperature":0.7,"pith_summary":"This paper sets out to prove that the partial sum process of sparsely observed, reconstructed random functions converges to its Gaussian limit in the Prokhorov metric at a polynomial rate, not merely weakly. The target is a finite-sample bound of the form $\\pi_\\infty(P_N, W) \\leq C N^{-\\tau}$ for a constant $\\tau > 0$, where $P_N$ is built from estimators $X_{i,M_N}$ that may be only approximately stationary and $W$ is a Brownian motion with the long-run covariance of the latent functions. If true, this is the first such rate for a function-valued partial sum process under $\\alpha$-mixing, and it yields new couplings between the process and Brownian motion, a strong invariance principle, and a validation of open-ended change-point monitoring for sparse functional data. The argument works by a two-step discretization: first replace the functional process by a high-dimensional grid approximation using entropy bounds, then apply a high-dimensional Gaussian approximation for weakly dependent vectors.","feed_headline":"Sparse functional sums approach Brownian motion at polynomial rate","feed_subtitle":"A polynomial Prokhorov bound makes open-ended monitoring valid for reconstructed functional curves.","key_machinery":"The proof's engine is a two-step discretization. First, for a grid $\\mathcal{G}_N$ on $[0,1] \\times [-N^\\rho, N^\\rho]$ with mesh $N^{-\\varsigma}$ and cardinality of order $N^{\\rho+2\\varsigma}$, the process $P_N$ is replaced by the step function $P_N^{\\mathrm{dis}}$ that reads off the value at the nearest gridpoint and is zero outside the growing cube; entropy and packing-number bounds together with a chaining moment inequality control the sup-norm difference between $P_N$ and $P_N^{\\mathrm{dis}}$, including the tail outside the cube. Second, because the discretized process lives in a finite-dimensional space of dimension $|\\mathcal{G}_N|$, a high-dimensional Gaussian approximation for $\\alpha$-mixing vectors (Lemma B.2, derived from Theorem 3.27 in [40]) gives a polynomial Prokhorov bound between $P_N^{\\mathrm{dis}}$ and a Gaussian with covariance $\\Sigma_N$, and a Wasserstein computation for Gaussian measures together with a matrix square-root proximity bound from [41] transfers to the Brownian covariance. A final step applies modulus-of-continuity and tail bounds for Banach-valued Brownian motion to undo the discretization.","core_discovery":"Theorem 4.1 is the central claim: under Assumptions 1 and 2 there are constants $C, \\tau > 0$ with $\\pi_\\infty(P_N, W) \\leq C N^{-\\tau}$. The metric is Prokhorov distance with respect to the supremum norm on $C_0(I_2, \\mathbb{R}^d)$, $P_N$ is the normalized partial sum process of reconstructed functions, and $W$ is the centered Brownian motion whose covariance is the long-run variance kernel of the latent functions restricted to $I_2$. The theorem covers fixed, slowly expanding, and infinite domains $I_2$, requires only polynomial $\\alpha$-mixing and approximate weak stationarity of the triangular array, and allows the reconstructed functions to be based on imbalanced samples. The authors read the bound as the first finite-sample, polynomial-rate distributional approximation between a functional partial sum process and its Brownian limit, and they build couplings, Wasserstein bounds, a bounded law of the iterated logarithm, an almost sure invariance principle under $\\beta$-mixing, and a monitoring validation on top of it.","pith_inferences":["The rate $\\tau$ is not made explicit and depends on hidden constants; an explicit version would require tracking all constants through the entropy, Gaussian approximation, and matrix perturbation steps, which the paper does not do.","The same discretize-then-approximate template should transfer to other Banach-valued partial sum processes, such as multivariate functional panels or functions on higher-dimensional cubes, whenever entropy bounds and a high-dimensional Gaussian approximation are available.","A concrete stress test is to verify Lemma B.2 against the original statement of Theorem 3.27 in [40] for dimension $|\\mathcal{G}_N|$ growing with $N$ and nonstationary rows; if that theorem requires fixed dimension or stationarity, the proof of Theorem 4.1 needs a different ingredient.","The monitoring application suggests a general principle: any polynomial Prokhorov bound on a growing compact domain can replace an infinite-domain strong approximation in validating open-ended sequential tests, which may extend to settings where KMT-type approximations are unavailable."],"forward_implications":["Corollary 1 yields a coupling of the reconstructed functions with $W$ such that $\\sup_{x\\in[0,1]} \\|P_N^{\\mathrm{lin}}(x) - W(x)\\| \\leq C N^{-\\tau}$ except on an event of probability at most $C N^{-\\tau}$.","Theorem 5.1 upgrades the coupling to $q$-th moment control for some $q > 2$, giving Wasserstein-distance convergence $W_q(P_N^{\\mathrm{lin}}, W) \\leq \\bar{C} N^{-\\bar{\\tau}}$ and an approximation of the $\\alpha$-mixing triangular array by independent Gaussian functions.","For fully observed functions, Corollary 2 gives a bounded law of the iterated logarithm under $\\alpha$-mixing and Corollary 3 gives an almost sure invariance principle with polynomial rate under $\\beta$-mixing.","Theorem 6.1 validates an open-ended CUSUM monitoring scheme for sparse functional time series: the test has asymptotic level $\\alpha$ and is consistent, using the coupling over a growing monitoring horizon instead of an infinite-domain strong approximation.","These consequences do not follow from ordinary weak convergence, because they require polynomial control of the Prokhorov distance rather than mere convergence in distribution."],"supporting_citations":[{"why":"Supplies Theorem 3.27, the high-dimensional Gaussian approximation for α-mixing vectors used in Lemma B.2 to prove Step 2 of Theorem 4.1.","marker":"[40]"},{"why":"Supplies Theorem 2.2.4 and packing-number bounds used to control the modulus of continuity of the linearized partial sum process and the discretization error in Lemma A.2.","marker":"[39]"},{"why":"Supplies Theorem 2.13 converting sup-norm closeness in probability into Prokhorov-metric closeness and back, used in Steps 1 and 3 and in Corollary 1.","marker":"[43]"},{"why":"Supplies Lemma 4.1 on the proximity of square roots of covariance matrices, used to bound the Wasserstein distance between the two Gaussian covariances in Step 2.","marker":"[41]"},{"why":"Supplies modulus-of-continuity estimates for Banach-valued Brownian motion used to control the distance between $W$ and its discretization in Step 3.","marker":"[42]"},{"why":"Supplies the blocking argument and moment bounds used to derive the almost sure invariance principle in Corollary 3 and the tail bound in the monitoring proof.","marker":"[21]"},{"why":"Supplies the univariate monitoring proof whose steps are adapted in Theorem 6.1 for the functional open-ended monitoring scheme.","marker":"[52]"},{"why":"Supplies moment inequalities for sums of strongly mixing random variables, used in Lemma A.2 and in the covariance estimates of Lemma A.3.","marker":"[64]"},{"why":"Supplies the inequality relating squared Prokhorov distance to Wasserstein distance, used in Step 2 and in Theorem 5.1.","marker":"[34]"}],"fun_headline_variants":["Polynomial Prokhorov bound for functional partial sums","Sparse functional curves converge to Brownian motion at polynomial rate","New coupling connects functional partial sums to Gaussian limits","Open-ended monitoring for sparse functional data validated"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The polynomial rate stands on an imported high-dimensional Gaussian approximation theorem that must apply to the specific discretized triangular array with dimension growing like $N^{\\rho+2\\varsigma}$; if that theorem does not cover nonstationary vectors of growing dimension, the bound in Theorem 4.1 is not proven.","fun_headline_variants_meta":{"raw":{"variants":["Polynomial Prokhorov bound for functional partial sums","Sparse functional curves converge to Brownian motion at polynomial rate","New coupling connects functional partial sums to Gaussian limits","Open-ended monitoring for sparse functional data validated"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000385,"raw_usage":{"total_tokens":2041,"prompt_tokens":956,"completion_tokens":1085,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":1022}},"tokens_in":572,"tokens_out":1085,"duration_ms":9382,"temperature":1.0,"reasoning_tokens":1022,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:34:20.659047+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the statement of Theorem 3.27 in [40] against the vectors $\\vec{v}_i$ defined in Lemma A.3: if it requires stationarity, fixed dimension, or stronger mixing than the paper's polynomial $\\alpha$-decay, then Lemma B.2 fails and Theorem 4.1 has no proof. Alternatively, simulate a simple stationary autoregressive functional process and estimate $\\pi_\\infty(P_N, W)$ for increasing $N$; an empirical decay slower than any polynomial rate would contradict the theorem.","supporting_citations":[{"cited_title":"Dehling, T","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 3.27, the high-dimensional Gaussian approximation for α-mixing vectors used in Lemma B.2 to prove Step 2 of Theorem 4.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 2.2.4 and packing-number bounds used to control the modulus of continuity of the linearized partial sum process and the discretization error in Lemma A.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 2.13 converting sup-norm closeness in probability into Prokhorov-metric closeness and back, used in Steps 1 and 3 and in Corollary 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 4.1 on the proximity of square roots of covariance matrices, used to bound the Wasserstein distance between the two Gaussian covariances in Step 2."},{"cited_title":"de Acosta, On the functional form of Levy’s modulus of continu- ity for Brownian motion, Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwandte Gebiete 69 (1985) 567–579","cited_arxiv_id":null,"evidence_quote":"Supplies modulus-of-continuity estimates for Banach-valued Brownian motion used to control the distance between $W$ and its discretization in Step 3."},{"cited_title":"Dehling, Limit theorems for sums of weakly dependent Banach space valued random variables, Z","cited_arxiv_id":null,"evidence_quote":"Supplies the blocking argument and moment bounds used to derive the almost sure invariance principle in Corollary 3 and the tail bound in the monitoring proof."},{"cited_title":"Horv´ ath, M","cited_arxiv_id":null,"evidence_quote":"Supplies the univariate monitoring proof whose steps are adapted in Theorem 6.1 for the functional open-ended monitoring scheme."},{"cited_title":"Yoshihara, Moment inequalities for mixing sequences, Kodai Math","cited_arxiv_id":null,"evidence_quote":"Supplies moment inequalities for sums of strongly mixing random variables, used in Lemma A.2 and in the covariance estimates of Lemma A.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the inequality relating squared Prokhorov distance to Wasserstein distance, used in Step 2 and in Theorem 5.1."}],"review_version":1}