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arxiv: 2605.20541 · v1 · pith:C5UOMCN4new · submitted 2026-05-19 · 🧮 math.ST · math.PR· stat.TH

Finite-Sample Bounds for Expected Signature Estimation under Weak Dependence

Bryson Schenck This is my paper

Pith reviewed 2026-05-21 06:17 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.TH
keywords expected signaturerough pathsfinite sample boundsweak dependenceblock averaging estimatorfractional Ornstein-Uhlenbeck processnon-asymptotic bounds
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The pith

Non-asymptotic mean-squared error bounds hold for block-averaging estimation of expected signatures under weak dependence.

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

This paper proves a finite-sample mean-squared error bound for estimating the expected signature of a random rough path from one long dependent trajectory using block averaging. The bound applies when paths are stationary with suitable moments and when block signatures satisfy a covariance decay condition weaker than alpha-mixing, allowing long-range dependence. The total error splits into a discretization term governed by the path's Hölder regularity and a fluctuation term governed by dependence strength. Explicit constants come from a level-by-level variance analysis that also yields an optimal allocation of blocks for a fixed number of observations. The assumptions are verified for fractional Ornstein-Uhlenbeck processes in rough, semimartingale, and long-memory regimes.

Core claim

We prove a non-asymptotic mean-squared error bound for the block-averaging estimator of the expected signature. Under moment and stationarity assumptions together with a covariance-decay condition on block signatures, the error separates into a discretization term and a fluctuation term, with rates determined respectively by path regularity and dependence strength. Rough-path theory ensures the signature is well-defined for paths with Hölder regularity at most 1/2. A level-wise rough-factorial variance analysis keeps finite-truncation constants explicit.

What carries the argument

The block-averaging estimator that partitions a single long trajectory into blocks and averages their signatures, controlled by a covariance-decay condition on those block signatures.

If this is right

  • The mean squared error decays at a rate determined by the minimum of the path regularity and the dependence decay rate.
  • An optimal block size exists that balances discretization and fluctuation errors for a fixed observation budget.
  • The bound applies directly to fractional Ornstein-Uhlenbeck processes with Hurst index in (0,1).
  • Empirical rates in simulations exceed the theoretical upper bounds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These bounds could support statistical inference procedures that use signatures to compare or classify dependent time series.
  • The approach might generalize to estimating other path functionals that require rough integration.
  • Practitioners could use the allocation rule to design experiments with limited data from long-memory processes.

Load-bearing premise

The block signatures must exhibit sufficient covariance decay to make the fluctuation term in the error bound vanish as the number of blocks grows.

What would settle it

Simulate many independent realizations of a fractional Ornstein-Uhlenbeck process with known parameters, compute the block-averaging estimator for increasing numbers of blocks, and check whether the observed mean squared error to the true expected signature follows the predicted decay rate.

Figures

Figures reproduced from arXiv: 2605.20541 by Bryson Schenck.

Figure 1
Figure 1. Figure 1: Log-log MSE plots for 𝐻 ∈ {0.40, 0.50, 0.60}. Dashed lines: panel-specific heuristic targets — (a) −𝛾 = −min(4𝐻 − 1, 2𝐻); (b) −𝜂; (c) −𝛾𝜂/(𝛾 + 𝜂). In these fOU experiments, the empirical convergence rates are consistent with — and strictly more negative than — the upper bounds of Theorem 3.3. 6. Discussion The non-asymptotic bounds of Theorem 3.3 decompose the estimation error into path-regularity and depe… view at source ↗
read the original abstract

The expected signature uniquely determines the law of a random rough path under a moment-growth condition, yet finite-sample bounds for estimating it from a single long dependent trajectory have been lacking. We study a stationary stochastic process whose sample paths can be interpreted as geometric rough paths, partitioned into blocks of equally-spaced observations, and prove a non-asymptotic mean-squared error bound for the block-averaging estimator. Rough-path theory is required for the estimand to be well-defined when paths have H\"older regularity at most $1/2$, because Young and Riemann--Stieltjes integration cannot define the signature's iterated integrals. Under moment and stationarity assumptions together with a covariance-decay condition on block signatures -- strictly weaker than $\alpha$-mixing and applicable to long-range-dependent drivers -- the error separates into a discretization term and a fluctuation term, with rates determined respectively by path regularity and dependence strength. A level-wise rough-factorial variance analysis keeps finite-truncation constants explicit and yields an optimal allocation rule under a fixed observation budget. We verify the assumptions for fractional Ornstein--Uhlenbeck processes in three regimes, namely rough (Hurst $H<1/2$), semimartingale ($H=1/2$), and long-range ($H>1/2$). Monte Carlo experiments show empirical convergence rates faster than the theoretical upper bounds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper establishes a non-asymptotic mean-squared error bound for a block-averaging estimator of the expected signature of a stationary process whose paths are interpreted as geometric rough paths. Under moment and stationarity assumptions plus a covariance-decay condition on successive block signatures (weaker than α-mixing and applicable to long-range dependence), the MSE decomposes into a discretization term governed by path Hölder regularity and a fluctuation term governed by dependence strength. A level-wise rough-factorial variance analysis keeps truncation constants explicit and yields an optimal block-size allocation under fixed observation budget. The assumptions are verified for fractional Ornstein–Uhlenbeck processes in the rough (H<1/2), semimartingale (H=1/2), and long-range (H>1/2) regimes; Monte Carlo experiments illustrate empirical rates faster than the derived bounds.

Significance. If the central bound and the verification for fractional OU processes hold, the work supplies the first explicit finite-sample guarantees for expected-signature estimation from a single long dependent trajectory, including under long-range dependence. The explicit constants, level-wise variance analysis, and optimal allocation rule are concrete strengths that make the result immediately usable for rough-path statistics and time-series applications with memory.

major comments (1)
  1. [§4.3] §4.3 (long-range regime verification): the covariance-decay condition on block signatures must be shown to produce a fluctuation term whose rate is strictly faster than the discretization term under the fixed-budget allocation; if the block-signature covariances inherit the power-law decay |i−j|^{2H−2} typical of LRD, the variance of the block average may only be O(N^{2H−2}), which would render the overall non-asymptotic bound non-informative for H>1/2. The manuscript states that the assumptions are verified, but an explicit decay-rate calculation for Cov(Sig(B_i), Sig(B_j)) is required to confirm the claimed separation of rates.
minor comments (2)
  1. [Definition 2.4] The statement of the covariance-decay condition (Definition 2.4 or equivalent) should include the precise exponent range that is admissible for the fluctuation term to dominate at the stated rate.
  2. [Figure 1] Figure 1 (Monte Carlo convergence plots): label the theoretical upper-bound curves explicitly so that the reader can see the gap between empirical and theoretical rates.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the long-range regime. We address the point directly below and will incorporate the requested clarification.

read point-by-point responses

  1. Referee: [§4.3] §4.3 (long-range regime verification): the covariance-decay condition on block signatures must be shown to produce a fluctuation term whose rate is strictly faster than the discretization term under the fixed-budget allocation; if the block-signature covariances inherit the power-law decay |i−j|^{2H−2} typical of LRD, the variance of the block average may only be O(N^{2H−2}), which would render the overall non-asymptotic bound non-informative for H>1/2. The manuscript states that the assumptions are verified, but an explicit decay-rate calculation for Cov(Sig(B_i), Sig(B_j)) is required to confirm the claimed separation of rates.

    Authors: We agree that an explicit decay-rate calculation strengthens the verification. The covariance-decay condition (Assumption 3.2) is satisfied by the fractional OU process for H > 1/2 because the block signatures are integrals over intervals of length n; the resulting Cov(Sig(B_i), Sig(B_j)) decays as |i−j|^{2H−2} multiplied by a factor that vanishes as n increases. Under the optimal allocation of block length n ∼ N^α with α chosen to balance the two error terms (as derived in Theorem 3.4), the fluctuation term is o(n^{−γ}) where γ > 0 is the Hölder-driven discretization exponent. Consequently the overall bound remains informative. We will insert the explicit computation of the block-signature covariance together with the resulting rate comparison into the revised §4.3. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct proof from stated assumptions

full rationale

The paper derives a non-asymptotic MSE bound for the block-averaging estimator by partitioning a stationary rough path into blocks and separating the error into a discretization term (controlled by Hölder regularity) and a fluctuation term (controlled by an explicit covariance-decay assumption on block signatures). This decay condition is introduced as a hypothesis strictly weaker than α-mixing and is verified separately for fractional OU processes in the H>1/2 regime; the bound itself follows from moment and stationarity assumptions without any reduction of the target quantity to a fitted parameter or self-referential prediction. No self-citations are load-bearing, no ansatz is smuggled, and the level-wise rough-factorial variance analysis keeps constants explicit under a fixed budget. The derivation is therefore self-contained against the listed assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central bound rests on standard domain assumptions from rough-path theory and stationary processes plus one paper-specific decay condition; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The process is stationary and satisfies moment-growth conditions that make the expected signature well-defined.
    Invoked to ensure the estimand exists and block averaging is unbiased in the limit.
  • domain assumption Covariance decay on block signatures is strictly weaker than alpha-mixing.
    Used to bound the fluctuation term; stated as applicable to long-range dependence.

pith-pipeline@v0.9.0 · 5768 in / 1364 out tokens · 43479 ms · 2026-05-21T06:17:04.903890+00:00 · methodology

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