Strong Gaussian approximations with random multipliers

Fabian Mies

REVIEW 2 major objections 5 minor 1 cited by

Strong Gaussian approximations with random multipliers

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that data-dependent random multipliers can be incorporated into strong Gaussian approximations for non-stationary, possibly high-dimensional time series, with explicit error rates and a functional central limit theorem as…

desk verdict Useful random-multiplier extension with a real proof gap in the de-randomization step of Theorem 3; the central result is likely correct but needs a genuine rework. read the letter →

arxiv 2412.14346 v1 pith:2WRKTBQY submitted 2024-12-18 math.ST stat.TH

Fabian Mies This is my paper · ORCID

classification math.STstat.TH MSC 60F1762M1060F0562L10

keywords Gaussianapproximationstrongfunctionalcentrallimittheoremlocallystationarytimeseriesrandommultipliershigh-dimensionalphysicaldependencesequentialtesting

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Classical functional central limit theorems fail for non-stationary or high-dimensional data because there is no single limit distribution to converge to. This paper proves that the partial sum process can still be uniformly approximated by a sequence of independent Gaussian vectors whose covariance matrices come from the time-varying local law, and that this approximation survives when every summand is multiplied by a data-dependent random matrix. The main theorem gives an explicit error bound in terms of the multiplier estimation error, the serial dependence tail, a delay parameter, and the growing dimensions. In the finite-dimensional locally stationary case the approximation upgrades to a functional central limit theorem, with an application to studentized sequential tests whose critical values do not depend on the unknown covariance.

What carries the argument

The proof rests on a block-martingale decomposition of the multiplier error. Write $\hat g_{t,n}=g_{t,n}+e_t$; divide the indices into blocks of length $L=L_n$, and in block $j$ replace $X_{t,n}$ by a copy $\tilde X_{t,n}$ built from fresh innovations. Assumption (A.M) makes $e_t$ measurable with respect to innovations at or before the block start, while $\tilde X_{t,n}$ uses later innovations, so $e_t$ and $\tilde X_{t,n}$ are independent inside the block; the block sums are then martingale differences whose $L^2$ norm is bounded by $\Theta_n\Lambda_n$. A second ingredient is the improved covariance estimate of Proposition 1, $\|\operatorname{Cov}(X_{s,n},X_{t,n})\|_{\mathrm{tr}}\le C_\beta(|s-t|+1)^{-\beta}$, which lets the Gaussian approximation run with only $\beta>1$ instead of $\beta>2$. The Gaussian vectors are then attached to the averaged process $g_{t,n}X_{t,n}$ via the strong-approximation theorem.

What would settle it

For iid standard normal $X_t$, let $\hat g_t=1+n^{-1/2}X_{t-1}$; the theorem predicts the normalized partial sum converges to a standard Brownian motion, and the correction term is $n^{-1}\sum X_{t-1}X_t\to0$. Replacing the multiplier by $\hat g_t=1+n^{-1/2}X_t$ (zero lag) makes the normalized partial sum process converge to $W_u+u$ rather than $W_u$, because $\frac1{\sqrt n}\sum_{t=1}^{\lfloor un\rfloor}\hat g_tX_t = \frac1{\sqrt n}\sum_{t=1}^{\lfloor un\rfloor}X_t + \frac1n\sum_{t=1}^{\lfloor un\rfloor}X_t^2 \Rightarrow W_u+u$; this concrete difference shows the positive lag in Assumption (A.M) is what prevents a bias term.

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Extended reading notes

Core claim

On a possibly enlarged probability space, for a non-stationary nonlinear time series $X_{t,n}=G_{t,n}(\epsilon_t)$ with zero mean and physical dependence coefficients decaying as $\Theta_n(h+1)^{-\beta}$ with $\beta>1$, and for random multiplier matrices $\hat g_{t,n}\in\mathbb{R}^{m\times d}$ that are measurable with respect to the innovations up to time $t-L_n$, one can construct independent Gaussian vectors $Y_{t,n}\sim N(0,\Sigma_{t,n})$ such that the maximum over $k\le n$ of $\|\frac{1}{\sqrt n}\sum_{t=1}^k(\hat g_{t,n}X_{t,n}-Y_{t,n})\|$ is stochastically bounded by the displayed rate. The rate contains a de-randomization error controlled by the $L^2$ distance $\Lambda_n$ between $\hat g_{t,n}$ and a deterministic approximation $g_{t,n}$, a dependence-tail term $\Xi_n=\sum_{h\ge L_n}\delta_n(h)$, and the Gaussian-approximation rate for $g_{t,n}X_{t,n}$. When the dimension is fixed and the sequence is locally stationary, Theorem 5 turns this into the functional convergence $\frac{1}{\sqrt n}\sum_{t=1}^{\lfloor un\rfloor}\hat g_{t,n}(X_{t,n}-E X_{t,n})\Rightarrow\int_0^u \Sigma_v^{1/2}\,dW_v$.

Load-bearing premise

The results stand or fall on Assumption (A.M): each multiplier at time $t$ may use only information up to time $t-L_n$ for some positive lag $L_n\ge1$, so that within each block the multiplier error is independent of a freshly randomized copy of the data; if $L_n=0$ the block argument collapses and the stated error bounds no longer follow.

Editorial extensions

If this is right

In a finite-dimensional locally stationary model, the studentized partial sum process $\frac1{\sqrt n}\sum_{t=1}^{\lfloor un\rfloor}\tilde\Sigma(t/n)^{-1/2}(X_{t,n}-E X_{t,n})$ converges to standard Brownian motion, so the critical values of the resulting sequential test do not depend on the unknown covariance function $\Sigma(u)$.
The error bound quantifies the price of data-dependent weights: the approximation degrades with the estimation error $\Lambda_n$, the dependence tail $\Xi_n$ beyond the lag, and the block length $L_n$.
Because $d$ and $m$ may grow with $n$, the result covers high-dimensional partial sums with high-dimensional multipliers, where no classical limit object exists.
The underlying Gaussian approximation is improved to any $\beta>1$, extending the earlier $\beta>2$ regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the same block-martingale argument should work for online estimators updated with a rolling window of length $L_n$ growing slowly, and the bound's $L_n\Phi_n n^{1/q-1/2}$ term gives the exact trade-off between adaptivity and accuracy.
Because the critical value in the studentized test is distribution-free, a practical extension is to turn the FCLT into a stopping rule for online monitoring: reject at the first $u$ where the studentized path crosses the Brownian supremum quantile, without knowing $\Sigma$ in advance.
For independent data the paper notes $\Xi_n=0$ for any $L_n\ge1$; this suggests the lag requirement is mainly a tool for dependence, and a version with contemporaneous multipliers may be possible for independent or conditionally independent arrays via different martingale arguments.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a genuinely useful extension of the strong approximation framework to random multipliers, and the studentization FCLT is a natural payoff. But the proof of Theorem 3 has a gap in the de-randomization step that is load-bearing, and the fix is not a one-line constant change.

What is actually new: Theorem 3 allows data-dependent multipliers with a lag, and Theorem 5 gives an FCLT for locally stationary sequences. Theorem 2 improves the beta range from beta>2 to beta>1 by exploiting Proposition 1's covariance bound. That is a real improvement. The application to sequential testing is coherent, and the simulations, while small, illustrate the power gain.

What is done well: The block-martingale idea is sensible, and the structure of the proof is clear. The paper is honest about relying on Mies-Steland (2023) for the base approximation, which is acceptable given that the target is an extension. The rate calculations in Theorem 2 are careful.

The soft spots: First, the proof of Theorem 3 asserts an equality for the maximal expectation of the martingale block sums, but only an inequality with a constant (Doob) is valid. That is minor and repairable. Second, and more serious, in equation (4) tilde X_t is coupled at the block boundary jL, so the lag from t to the resampled innovation is d = t - jL, ranging from 1 to L. The L2 error for each term is of order Theta_n d^{1-beta}, not Theta_n L^{1-beta}. Summing squares over blocks gives roughly (n/L) sum_{d=1}^L d^{2-2beta} ~ n/L for beta>3/2, which is larger than the claimed n L^{2-2beta} by a factor L^{2beta-3}. So the displayed OP(Lambda_n) OP(sqrt(n) Xi_n) bound for the residual does not follow from the construction as written. That matters because the de-randomization term Lambda_n Xi_n feeds into Theorem 3 and into the hypotheses of Theorem 5. The natural fix—resampling at lag L for each t—would break the martingale difference structure of the first term, so the repair is not trivial. The theorem may well be true with a moderately worse rate, but it needs a real argument.

There are also typos: a B/L mismatch in the residual term, an incorrect exponent k_n ~ n^{-1/3} in the proof of Theorem 6 (should be n^{2/3}), and a line saying Theta_n = 0 where the context indicates Xi_n = 0.

Bottom line: the paper deserves a serious referee, but the referee should require a corrected proof of Theorem 3. The central contribution is real and likely correct; it just isn't fully proved as written.

Referee Report

2 major / 5 minor

Summary. The paper extends strong Gaussian approximation results for partial sums of non-stationary, potentially high-dimensional time series to sums whose summands are multiplied by data-dependent random matrices. The main result (Theorem 3) gives an explicit coupling between the multiplier-weighted partial sum process and a sequence of independent Gaussian vectors, with an error rate expressed in terms of the multiplier estimation error (Λ_n), the dependence tail (Ξ_n), the multiplier regularity (Φ_n, Ψ_n), and a block-length term. In the finite-dimensional locally stationary setting (Theorem 5), the coupling yields a functional central limit theorem. An application to studentized partial sums and sequential testing is given in Theorem 6, with a small simulation study. The paper also improves the underlying strong approximation (Theorem 2) by weakening the dependence condition from β > 2 to β > 1. The central claims are plausible and the rates are clearly stated, but the proof of the de-randomization step in Theorem 3 currently contains a load-bearing gap, in addition to a false equality in a maximal inequality display.

Significance. If the results are correct, they constitute a useful extension of strong Gaussian approximations to settings with estimated or data-dependent multipliers, which is relevant for bootstrap and sequential inference in non-stationary environments. The explicit treatment of the lagged measurability assumption (A.M) and the resulting Ξ_n terms is a genuine contribution, and the improved β > 1 condition in Theorem 2 over the prior work of Mies and Steland is valuable. The paper ships a concrete statistical application (studentized partial sums) with simulation evidence, and the main theorem is stated with explicit rates. These are real strengths. However, the proof of Theorem 3 is not complete as written: one equality is invalid, and the residual bound for the de-randomization step relies on a coupling that does not produce the stated rate. Both issues appear repairable without changing the stated rates, but they are central to the main theorem.

major comments (2)

[Section 3, Proof of Theorem 3, display after Eq. (4)] The identity E max_{r} ||∑_{j=0}^{r} W_j||^2 = ∑_{j} E||W_j||^2 is not valid for the block martingale differences W_j; only an inequality, e.g. Doob's L2 maximal inequality, gives E max_r ||S_r||^2 ≤ C ∑_j E||W_j||^2. Since the subsequent computation bounds exactly the sum of block second moments, the final rate is unchanged up to a universal constant, but the displayed equality must be replaced by an inequality.
[Section 3, Proof of Theorem 3, second term in Eq. (4)] The coupling \tilde X_{t,n} = G_{t,n}(\tilde ε^j_{t,t-jL}) replaces innovations at indices ≤ jL, so for t = jL + d with 1 ≤ d ≤ L, the residual X_{t,n} - \tilde X_{t,n} depends on innovations at lags at least d. By (A.2), ||X_{t,n} - \tilde X_{t,n}||_{L2} is of order Θ_n d^{1-β}. Consequently, ∑_t ||X_{t,n} - \tilde X_{t,n}||_{L2}^2 is of order (n/L) ∑_{d=1}^L d^{2-2β}, which for β > 3/2 is of order n/L, whereas (√n Ξ_n)^2 is of order n L^{2-2β} and is strictly smaller when β > 3/2. Hence the displayed bound OP(Λ_n) OP(√n Ξ_n) for the second term in (4) does not follow from the construction used. This gap is load-bearing because the rate Λ_n Ξ_n enters Theorem 3 and is one of the conditions for Theorem 5. The proof should be repaired, for example by resampling at a fixed lag L in a way that still preserves the block martingale-difference structure, so that the residual tail starts at L.

minor comments (5)

[Section 4, Theorem 5] The statement should explicitly require L_n ≥ 1 (or L_n ∈ N) in addition to L_n = o(n^{1/2 - 1/q}), so that Theorem 3 applies; the current condition alone does not guarantee the lagged measurability needed for the de-randomization argument.
[Section 5, Proof of Theorem 6] The sentence 'with L_n = 1 and Θ_n = 0' should read 'with L_n = 1 and Ξ_n = 0': the dependence constant Θ_n is positive even for independent data, while the tail sum Ξ_n vanishes for any L_n ≥ 1.
[Section 5, Proof of Theorem 6] The statement 'for k_n ≍ n^{-1/3}' is a typo; it should be 'k_n ≍ n^{2/3}' (or a positive power consistent with k_n = \lfloor n^{2/3} \rfloor) to match the subsequent Λ_n = O(n^{1/6}).
[Section 3, after Theorem 3] The phrase 'Assumption (A.4) implies' should refer to Assumption (A.M), not (A.4), which is defined later in Section 4.
[Section 2, notation for ε_{t,h} and \tilde ε_{t,h}] The notation for the coupled innovation sequences is confusing: \tilde ε_{t,h} and ε_{t,h} are defined with different replacement schemes, and the proof of Theorem 3 uses \tilde ε^j_{t,t-jL} but the explanatory formula in the text appears to describe ε_{t,h}. The authors should clarify which sequence is used in the block coupling.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the random-multiplier approximation is derived from an externally published base theorem and does not assume its own conclusion.

full rationale

The central derivation chain is not circular. Theorem 3 reduces the target random-multiplier statement to (i) a de-randomization step that bounds the effect of replacing the random multiplier \hat g_{t,n} by the non-random g_{t,n} and (ii) the deterministic-multiplier strong approximation Theorem 2, applied to Z_{t,n}=g_{t,n}G_{t,n}(\epsilon_t). Theorem 2 is not assumed as the target result; it is an improved version of a previously published theorem (Mies and Steland 2023), and the improvement is justified in the paper via Proposition 1, which is proved from assumption (A.2). Thus the random-multiplier result has independent content beyond the prior theorem. No parameters are fitted and renamed as predictions; the rates are derived from the stated constants Λ_n, Ξ_n, Φ_n, Ψ_n, and the physical-dependence decay. The paper also explicitly notes that the FCLT without multipliers is standard in the literature, so Theorem 5 is not a renaming of a known result. The proof of Theorem 3 contains a potentially serious gap: the bound on the second term in decomposition (4) is asserted as OP(Λ_n)OP(√n Ξ_n), whereas the blockwise coupling \tilde X_t=G_{t,n}(\tilde\epsilon^j_{t,t-jL}) gives an L2 sum of order n/L, which for β>3/2 is larger than n Ξ_n^2; this is a correctness concern, not a circularity, and there is no reduction of the theorem to its own assumptions.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted: all constants are universal or depend only on beta and q. The paper introduces no new unobserved entities. The central claim rests on the physical dependence framework, the polynomial decay bound (A.2), the lagged measurability (A.M), and the prior strong approximation of Mies and Steland (2023). The proof gap around the maximal inequality is flagged but fixable without changing the rates.

assumptions (7)

standard math Physical dependence representation: X_{t,n}=G_{t,n}(epsilon_t) with iid seeds and martingale-difference projection (Wu, 2005).
Used throughout, especially in the proof of Proposition 1, to bound covariances and dependence. This is a standard tool in nonlinear time series theory.
domain assumption (A.1) Time smoothness: ||G_{1,n}(epsilon_0)||_{L2} + sum_{t=2}^n ||G_{t,n}(epsilon_0)-G_{t-1,n}(epsilon_0)||_{L2} <= Theta_n Gamma_n.
Controls the drift of the data generating mechanism over time and feeds into the approximation error in Theorems 2 and 3.
domain assumption (A.2) Polynomial decay of physical dependence: delta_n(h) <= Theta_n (h+1)^{-beta} with beta > 1.
Ensures the local long-run covariance Sigma_{t,n} is finite; Proposition 1 derives the trace covariance bound from it. The paper's rate improvement is to weaken beta > 2 to beta > 1.
domain assumption (A.M) Lagged measurability: ghat_t is epsilon_{t-L_n}-measurable.
Essential for the block decomposition in Theorem 3; without a positive lag, e_t and tilde X_t need not be independent and the de-randomization error bound fails.
domain assumption (A.3) and (A.4) Local stationarity: maps G_u and g_u approximate G_{floor(vn),n} and g_{floor(vn),n} in L2 as n tends to infinity.
Used only in Section 4 to obtain the FCLT via Lemma 4; not needed for the high-dimensional approximation theorems.
domain assumption The base strong approximation theorem of Mies and Steland (2023, Thm 3.1) and its proof skeleton.
Theorem 2 is presented as an improvement of that theorem, and Theorem 3 applies Theorem 2 to g_{t,n} X_{t,n}. The present paper does not reprove the base result from scratch.
standard math L2 martingale maximal inequality for vector-valued martingale difference sequences.
Needed to pass from block variances to the maximum over blocks; the proof currently writes this as an equality, but only an inequality with a constant is available.

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Pith. "Pith review of Strong Gaussian approximations with random multipliers." pith.science (2026). https://pith.science/paper/2WRKTBQY

@misc{pith2026241214346,
  author       = {Pith},
  title        = {Pith review of: Strong Gaussian approximations with random multipliers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2WRKTBQY}},
  note         = {Machine review of arXiv:2412.14346}
}

read the original abstract

One reason why standard formulations of the central limit theorems are not applicable in high-dimensional and non-stationary regimes is the lack of a suitable limit object. Instead, suitable distributional approximations can be used, where the approximating object is not constant, but a sequence as well. We extend Gaussian approximation results for the partial sum process by allowing each summand to be multiplied by a data-dependent matrix. The results allow for serial dependence of the data, and for high-dimensionality of both the data and the multipliers. In the finite-dimensional and locally-stationary setting, we obtain a functional central limit theorem as a direct consequence. An application to sequential testing in non-stationary environments is described.

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Berkes, I., Liu, W., and Wu, W. B. (2014). Koml´ os-Major-Tusn´ ady approximation under dependence. Annals of Probability, 42(2):794–817

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Billingsley, P. (1999). Convergence of Probability Measures. John Wiley & Sons

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work page 2024

[4] [4]

Dahlhaus, R. (1997). Fitting time series models to nonstationary pr ocesses. Annals of Statistics, 25(1):1–37

work page 1997

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and Polonik, W

Dahlhaus, R. and Polonik, W. (2009). Empirical spectral processe s for locally stationary time series. Bernoulli, 15(1):1–39

work page 2009

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Dahlhaus, R., Richter, S., and Wu, W. B. (2019). Towards a general theory for nonlinear locally stationary processes. Bernoulli, 25(2):1013–1044

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