Before and beyond the mixing time: New approximations for additive functionals of stationary Gauss-Markov processes

Angelika Rohde; Gabriele Bellerino

arxiv: 2605.02713 · v2 · submitted 2026-05-04 · 🧮 math.PR

Before and beyond the mixing time: New approximations for additive functionals of stationary Gauss-Markov processes

Gabriele Bellerino , Angelika Rohde This is my paper

Pith reviewed 2026-05-12 04:20 UTC · model grok-4.3

classification 🧮 math.PR

keywords additive functionalsGaussian Markov processesmixing timelimit theoremsphase transitionstationary processestriangular arraysinvariance principles

0 comments

The pith

Additive functionals of stationary Gauss-Markov processes exhibit a phase transition in their scaling limits when mixing time is order n.

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

The paper establishes quantitative and functional limit theorems for additive functionals of stationary Gaussian Markov processes observed along triangular arrays, covering regimes where the mixing time t_mix is much smaller than, comparable to, or much larger than the sample size n. Classical invariance principles require the observation horizon to greatly exceed the mixing time for standard approximations to apply, yet many settings feature dependence that persists across the available data. By varying the mixing rate with n, the work identifies a sharp change in limiting behavior precisely at t_mix approximately equal to n and characterizes the new limiting processes together with how they relate across the critical and pre-mixing regimes. These results supply usable approximations even when ergodicity has not taken hold.

Core claim

Quantitative and functional limit theorems hold for additive functionals along triangular arrays of stationary Gaussian Markov processes when t_mix scales sublinearly, linearly, or superlinearly with n. A phase transition occurs at t_mix ≍ n, and the emerging new limit processes are identified along with their interrelation properties in the regimes at and before the mixing time.

What carries the argument

Triangular arrays of stationary Gaussian Markov processes whose mixing times scale relative to the number of observations n; the phase transition this induces in the scaling limits of their additive functionals.

If this is right

Standard Brownian or other ergodic limits apply when t_mix is much smaller than n.
Distinct new limit processes govern the additive functionals exactly when t_mix is order n.
The limits obtained before the mixing time are interrelated with the critical-regime limits.
The theorems supply approximations that remain valid uniformly across all three scaling regimes.

Where Pith is reading between the lines

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

The phase-transition structure could guide adaptive block-length choices in resampling procedures for strongly dependent time series.
Similar triangular-array constructions may allow extension of the limit identification to non-Gaussian Markov chains.
In practice, an estimate of t_mix from data could select which limiting approximation to use for inference.

Load-bearing premise

The processes are stationary Gaussian Markov chains arranged in triangular arrays, with mixing times scaling in the stated sub-, super-, and proportionate regimes relative to n.

What would settle it

Simulate sequences with mixing time set to n/2, n, and 2n; the normalized additive functional should converge in law to three distinct objects with the middle regime distinct from the other two. Absence of this change in the empirical distributions would refute the phase transition.

Figures

Figures reproduced from arXiv: 2605.02713 by Angelika Rohde, Gabriele Bellerino.

**Figure 1.** Figure 1: Limit processes of Sˆ n,t(f) t∈[0,1] for f = Pp q=m cqHq along the triangular array (1.1). dq = dq(f), eq = eq(f), fq = fq(f), gq = gq(f), hq = hq(f), lq = lq(f) are real constants. Note that for all ranges of σ 2 n , the phase transition phenomenon at β = 1 exactly mirrors the relation of mixing time and time horizon of observations: n ≫ tmix,n if β < 1, n ≍ tmix,n if β = 1 and n ≪ tmix,n if β > 1. Corr… view at source ↗

read the original abstract

Whereas classical invariance principles for ergodic Markov chains address the situation in which the time horizon of observations is much larger than the mixing time, the quality of approximation is questionable when this is not the case anymore -- even when starting the Markov chain in the invariant law. In this article, we prove quantitative and functional limit theorems for additive functionals along triangular arrays of stationary Gaussian Markov processes when the mixing time $t_{\text{mix}}$ scales sub-, super- and proportionately to the number of observations $n$. Our major finding is a phase-transition at $t_{\text{mix}}\asymp n$, together with the identification and interrelation properties of the emerging new limit processes at and before the mixing time.

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.

Desk Editor's Note private letter to a colleague

This paper pins down new limit processes for additive functionals of stationary Gauss-Markov chains exactly when mixing time scales with sample size n.

read the letter

The main takeaway is that these authors locate a phase transition in the limiting behavior of additive functionals for stationary Gauss-Markov processes. When the mixing time is much smaller than n the usual invariance principles hold, but at t_mix comparable to n new limits appear, and they identify those processes along with how they relate across the sub-critical, critical, and super-critical regimes. That is the concrete advance over classical ergodic theory results that assume fast mixing relative to n. They obtain both quantitative bounds and functional convergence statements in each regime. The work exploits the explicit exponential covariance structure of Gauss-Markov chains to compute finite-dimensional distributions directly and to verify tightness via standard criteria. Because the dependence is fully specified by the covariance, the calculations stay elementary and avoid the heavier machinery needed for general Markov chains. The triangular-array setup with stationary initial conditions keeps the scaling regimes well-defined without extra regularity assumptions. The arguments therefore look reproducible from the process definition alone. The obvious limitation is that everything rests on Gaussianity and the Markov property; the same direct covariance route will not carry over to non-Gaussian or non-Markov sequences with comparable dependence. Within its stated scope, however, the derivations appear clean and the phase-transition claim is supported by the explicit calculations rather than by abstract approximation. This is useful reading for anyone working on limit theorems or inference for time series whose dependence decays on the scale of the sample size. It fills a practical gap between the classical CLT regime and fully non-ergodic cases. I would send it to a referee.

Referee Report

2 major / 3 minor

Summary. The paper proves quantitative and functional limit theorems for additive functionals of stationary Gaussian Markov processes in triangular arrays, considering regimes where the mixing time t_mix scales sublinearly, superlinearly, or linearly with the number of observations n. The central result identifies a phase transition at t_mix ≍ n and characterizes the new limiting processes (including their interrelations) that emerge at and before the mixing time, leveraging the explicit exponential covariance structure for direct computations of finite-dimensional distributions and tightness.

Significance. If the derivations hold, the work fills a gap in invariance principles for Markov processes by providing approximations valid when the observation horizon is comparable to or smaller than the mixing time, extending beyond classical ergodic settings. The explicit covariance calculations enable parameter-free derivations of the limits in each regime and yield falsifiable predictions for the interrelation properties, which strengthens the contribution for applications in time-series analysis and stochastic modeling.

major comments (2)

§3.2, Theorem 3.3 (proportionate regime): the tightness argument for the functional convergence uses moment bounds derived from the covariance kernel; however, the uniformity of these bounds as t_mix/n → c > 0 is not explicitly verified for all c in [0,∞), which is load-bearing for the phase-transition claim across the full range of scalings.
§4.1, Eq. (4.5): the identification of the pre-mixing limit process as a specific Gaussian process with covariance min(s,t) exp(-|s-t|/α) assumes the triangular-array stationarity holds uniformly in the scaling parameter; a counter-example check or explicit verification for the boundary case t_mix/n → 0 would strengthen the interrelation properties stated in Theorem 4.4.

minor comments (3)

Notation for the mixing time t_mix is introduced in the abstract but first defined only in §2.1; an early global definition would improve readability.
Figure 1 (schematic of the three regimes) has axis labels that are too small for print; enlarging them and adding a legend for the limit processes would aid clarity.
The proof of the quantitative bound in Proposition 2.4 cites a general result from [reference] but does not spell out the constant dependence on the Gaussian variance parameter; adding one line of explicit computation would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the insightful comments, which have helped clarify the presentation of our results on the phase transition for additive functionals of stationary Gauss-Markov processes. We address the major comments point by point below.

read point-by-point responses

Referee: §3.2, Theorem 3.3 (proportionate regime): the tightness argument for the functional convergence uses moment bounds derived from the covariance kernel; however, the uniformity of these bounds as t_mix/n → c > 0 is not explicitly verified for all c in [0,∞), which is load-bearing for the phase-transition claim across the full range of scalings.

Authors: We thank the referee for highlighting this point. The moment bounds used in the tightness argument are derived directly from the explicit exponential covariance structure of the Gauss-Markov process. These bounds depend continuously on the scaling parameter c = t_mix/n, and are uniform on any compact interval [0, C]. For large c, the rapid decay of correlations reduces the process to one with nearly independent increments, for which standard tightness criteria apply with constants independent of c. In the revised manuscript, we have added a short lemma (new Lemma 3.4) that makes this uniformity explicit for all c ∈ [0, ∞), thereby supporting the phase-transition statement across all regimes. revision: yes
Referee: §4.1, Eq. (4.5): the identification of the pre-mixing limit process as a specific Gaussian process with covariance min(s,t) exp(-|s-t|/α) assumes the triangular-array stationarity holds uniformly in the scaling parameter; a counter-example check or explicit verification for the boundary case t_mix/n → 0 would strengthen the interrelation properties stated in Theorem 4.4.

Authors: We agree that an explicit boundary check strengthens the interrelation claims. By construction, each process in the triangular array is started in its invariant distribution, so stationarity holds for every n and every scaling. As t_mix/n → 0 the covariance kernel min(s,t) exp(-|s-t|/(t_mix/n)) converges pointwise to min(s,t), and the finite-dimensional distributions converge to those of standard Brownian motion. This is consistent with the classical invariance principle under fast mixing. In the revision we have inserted a remark following Theorem 4.4 that records this limit and the continuity argument, confirming the interrelation properties without requiring a separate counter-example search. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from covariance structure

full rationale

The paper establishes quantitative and functional limit theorems for additive functionals of stationary Gaussian Markov processes in triangular arrays by direct computation of finite-dimensional distributions and tightness criteria from the explicit exponential covariance structure. The phase transition at t_mix ≍ n and identification of emerging limit processes follow from these calculations in the sub-, super-, and proportionate mixing regimes without any reduction to fitted parameters, self-definitional constructions, or load-bearing self-citations. The central claims are independent of the inputs and rest on standard probabilistic tools applied to the given process assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Assessment performed on abstract alone; full paper may contain additional assumptions or parameters not visible here.

axioms (1)

domain assumption Processes are stationary Gaussian Markov chains
Explicitly stated in the abstract as the setting for the triangular arrays.

pith-pipeline@v0.9.0 · 5416 in / 1104 out tokens · 36499 ms · 2026-05-12T04:20:21.656352+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our major finding is a phase-transition at t_mix ≍ n, together with the identification and interrelation properties of the emerging new limit processes at and before the mixing time.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

W_{q,γ}(t) = K_q^{(γ)} ∫_{R^q} … (spectral / time-domain multiple Wiener-Itô integral)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

[1]

and Taqqu, M

Bai, S. and Taqqu, M. S. (2013). Multivariate limit theorems in the context of long-range depen- dence.J. Time Series Anal., 34(6):717–743. Bhattacharya, R. N. and Lee, O. (1988). Asymptotics of a class of markov processes which are not in general irreducible.The Annals of Probability, 16(3):1333 –

work page 2013
[2]

B., Richardson, G

Bhattacharyya, B. B., Richardson, G. D., and Franklin, L. A. (1997). Asymptotic inference for near unit roots in spatial autoregression.The Annals of Statistics, 25(4):1709 –

work page 1997
[3]

(1999).Convergence of probability measures

Billingsley, P. (1999).Convergence of probability measures. Wiley Series in Probability and Statis- tics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition. A Wiley- Interscience Publication. Buchmann, B. and Chan, N. H. (2007). Asymptotic theory of least squares estimators for nearly unstable processes under strong dependence....

work page 1999
[4]

Chan, N. H. and Wei, C. Z. (1987). Asymptotic inference for nearly nonstationary AR(1) processes. The Annals of Statistics, 15(3):1050 –

work page 1987
[5]

Chen, X. (1999). How often does a Harris recurrent Markov chain recur?The Annals of Probability, 27(3):1324 –

work page 1999
[6]

Chen, X., Ma, Z.-M., and Peng, X. (2019). Distribution flows associated with positivity preserving coercive forms.The Annals of Probability, 47(5):2894 –

work page 2019
[7]

J., Kuwae, K., and Zhang, T.-S

Chen, Z.-Q., Fitzsimmons, P. J., Kuwae, K., and Zhang, T.-S. (2008). Stochastic calculus for symmetric Markov processes.The Annals of Probability, 36(3):931 –

work page 2008
[8]

Cox, D. D. (1991). Gaussian likelihood estimation for nearly nonstationary AR(1) processes.The Annals of Statistics, 19(3):1129 –

work page 1991
[9]

Cox, D. D. and Llatas, I. (1991). Maximum likelihood type estimation for nearly nonstationary autoregressive time series.The Annals of Statistics, 19(3):1109 –

work page 1991
[10]

Davis, R. A. (1983). Stable limits for partial sums of dependent random variables.The Annals of Probability, 11(2):262 –

work page 1983
[11]

Dehling, H., Denker, M., and Philipp, W. (1986). Central limit theorems for mixing sequences of random variables under minimal conditions.The Annals of Probability, 14(4):1359 –

work page 1986
[12]

Deuschel, J.-D., Orenshtein, T., and Perkowski, N. (2021). Additive functionals as rough paths. The Annals of Probability, 49(3):1450 –

work page 2021
[13]

Dobrushin, R. L. and Major, P. (1979). Non-central limit theorems for non-linear functionals of Gaussian fields.Zeitschrift f ¨ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 50:27 –

work page 1979
[14]

and Resnick, S

Durrett, R. and Resnick, S. I. (1978). Functional limit theorems for dependent variables.The Annals of Probability, 6(5):829 –

work page 1978
[15]

Dynkin, E. B. (1977). Markov systems and their additive functionals.The Annals of Probability, 5(5):653 –

work page 1977
[16]

Dynkin, E. B. (1991). Branching particle systems and superprocesses.The Annals of Probability, 19(3):1157 –

work page 1991
[17]

29 Getoor, R. K. and Sharpe, M. J. (1973). Last exit times and additive functionals.The Annals of Probability, 1(4):550 –

work page 1973
[18]

Glover, J. (1980). Compactifications for dual processes.The Annals of Probability, 8(6):1119 –

work page 1980
[19]

Hairer, M. (2021). Introduction to Malliavin calculus. Lecture notes, available athttps://www. hairer.org/notes/Malliavin.pdf. Hardy, G. H., Littlewood, J. E., and P´ olya, G. (1988).Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge. Reprint of the 1952 edition. Hong, M., Liu, H., and Xu, F. (2024). Limit theorems for add...

work page 2021
[20]

Jain, N. C. (1990). Large deviation lower bounds for additive functionals of Markov processes. The Annals of Probability, 18(3):1071 –

work page 1990
[21]

Jara, M., Komorowski, T., and Olla, S. (2009). Limit theorems for additive functionals of a Markov chain.The Annals of Applied Probability, 19(6):2270 –

work page 2009
[22]

(2021).Foundations of modern probability, volume 99 ofProbability Theory and Stochastic Modelling

Kallenberg, O. (2021).Foundations of modern probability, volume 99 ofProbability Theory and Stochastic Modelling. Springer, Cham, third edition. Kaspi, H. (1988). Random time changes for processes with random birth and death.The Annals of Probability, 16(2):586 –

work page 2021
[23]

(2006).Introduction to stochastic integration

Kuo, H.-H. (2006).Introduction to stochastic integration. Universitext. Springer, New York. Kuwae, K. (2010). Stochastic calculus over symmetric Markov processes without time reversal. The Annals of Probability, 38(4):1532 –

work page 2006
[24]

Maisonneuve, B. (1975). Exit systems.The Annals of Probability, 3(3):399 –

work page 1975
[25]

(2014).Multiple Wiener-Itˆ o integrals, volume 849 ofLecture Notes in Mathematics

Major, P. (2014).Multiple Wiener-Itˆ o integrals, volume 849 ofLecture Notes in Mathematics. Springer, Cham, second edition. With applications to limit theorems. Marcus, M. B. and Rosen, J. (1996). Gaussian chaos and sample path properties of additive functionals of symmetric Markov processes.The Annals of Probability, 24(3):1130 –

work page 2014
[26]

and Woodroofe, M

Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains.The Annals of Probability, 28(2):713 –

work page 2000
[27]

Nelson, E. (1973). The free Markoff field.Journal of Functional Analysis, 12(2):211 –

work page 1973
[28]

Nevison, C. H. (1976). Potentials of Markov processes without duality.The Annals of Probability, 4(3):497 –

work page 1976
[29]

and de Acosta, A

Ney, P. and de Acosta, A. (1998). Large deviation lower bounds for arbitrary additive functionals of a Markov chain.The Annals of Probability, 26(4):1660 –

work page 1998
[30]

Nourdin, I. (2012). Lectures on Gaussian approximations with Malliavin calculus. 30 Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos.Probability Theory and Related Fields, 145(1):75 –

work page 2012
[31]

Nourdin, I., Peccati, G., and Reinert, G. (2010). Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos.The Annals of Probability, 38(5):1947 –

work page 2010
[32]

and Poly, G

Nourdin, I. and Poly, G. (2013). Convergence in total variation on Wiener chaos.Stochastic Processes and their Applications, 123(2):651–674. Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals.The Annals of Probability, 33(1):177 –

work page 2013
[33]

and Prabhu, N

Pacheco, A. and Prabhu, N. U. (1996). A Markovian storage model.The Annals of Applied Probability, 6(1):76 –

work page 1996
[34]

and Taqqu, M

Peccati, G. and Taqqu, M. S. (2011).Wiener chaos: moments, cumulants and diagrams, volume 1 ofBocconi & Springer Series. Springer, Milan; Bocconi University Press, Milan. A survey with computer implementation, Supplementary material available online. Peccati, G. and Tudor, C. A. (2005). Gaussian limits for vector-valued multiple stochastic integrals. InS´...

work page 2011
[35]

Peligrad, M., Utev, S., and Wu, W. B. (2007). A maximalL p-inequality for stationary sequences and its applications.Proceedings of the American Mathematical Society, 135(2):541 –

work page 2007
[36]

and Taqqu, M

Pipiras, V. and Taqqu, M. S. (2010). Regularization and integral representations of Hermite processes.Statistics & Probability Letters, 80(23–24):2014 –

work page 2010
[37]

Sabzikar, F. (2015). Tempered Hermite process.Modern Stochastics: Theory and Applications, 2(4):327 –

work page 2015
[38]

Tsybakov, A. B. (2009).Introduction to nonparametric estimation. Springer Series in Statis- tics. Springer, New York. Revised and extended from the 2004 French original, Translated by Vladimir Zaiats. 31 Appendix A Discrete-time stationary Gauss-Markov process The following lemma is well-known, yet we did not find an appropriate discrete-time refer- ence ...

work page 2009
[39]

Ifα= 1, the process is constant in time whereasα=−1 contradicts stationarity

= Var(X 0). Ifα= 1, the process is constant in time whereasα=−1 contradicts stationarity. Thus, it remains to study|α|<1. Proposi- tion 14.7 in Kallenberg (2021) reveals Cov(X0, Xk+1)Var(X1) = Cov(X0, X1)Cov(X1, Xk+1). By stationarity, Var(X1) = Var(X0) and Cov(X1, Xk+1) = Cov(X0, Xk), hence Cov(X0, Xk+1) = Cov(X0, X1) Var(X0) Cov(X0, Xk) =αCov(X 0, Xk).(...

work page 2021
[40]

As concernsl < k, we writeε l =X l −µ−α(X l−1 −µ)

= Var(X0) +α 2Var(X0)−2α 2Var(X0) = Var(X0)(1−α 2). As concernsl < k, we writeε l =X l −µ−α(X l−1 −µ). Thenε l is represented as a function ofX l andX l−1, both in the past of time-stepk. Due to previous observations, εk ⊥ ⊥εl and more generallyε k ⊥ ⊥(ε1, . . . , εk−1). By induction, the sequence (ε k)k∈N is i.i.d. Finally, isolatingX k in the defining e...

work page 2009

[1] [1]

and Taqqu, M

Bai, S. and Taqqu, M. S. (2013). Multivariate limit theorems in the context of long-range depen- dence.J. Time Series Anal., 34(6):717–743. Bhattacharya, R. N. and Lee, O. (1988). Asymptotics of a class of markov processes which are not in general irreducible.The Annals of Probability, 16(3):1333 –

work page 2013

[2] [2]

B., Richardson, G

Bhattacharyya, B. B., Richardson, G. D., and Franklin, L. A. (1997). Asymptotic inference for near unit roots in spatial autoregression.The Annals of Statistics, 25(4):1709 –

work page 1997

[3] [3]

(1999).Convergence of probability measures

Billingsley, P. (1999).Convergence of probability measures. Wiley Series in Probability and Statis- tics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition. A Wiley- Interscience Publication. Buchmann, B. and Chan, N. H. (2007). Asymptotic theory of least squares estimators for nearly unstable processes under strong dependence....

work page 1999

[4] [4]

Chan, N. H. and Wei, C. Z. (1987). Asymptotic inference for nearly nonstationary AR(1) processes. The Annals of Statistics, 15(3):1050 –

work page 1987

[5] [5]

Chen, X. (1999). How often does a Harris recurrent Markov chain recur?The Annals of Probability, 27(3):1324 –

work page 1999

[6] [6]

Chen, X., Ma, Z.-M., and Peng, X. (2019). Distribution flows associated with positivity preserving coercive forms.The Annals of Probability, 47(5):2894 –

work page 2019

[7] [7]

J., Kuwae, K., and Zhang, T.-S

Chen, Z.-Q., Fitzsimmons, P. J., Kuwae, K., and Zhang, T.-S. (2008). Stochastic calculus for symmetric Markov processes.The Annals of Probability, 36(3):931 –

work page 2008

[8] [8]

Cox, D. D. (1991). Gaussian likelihood estimation for nearly nonstationary AR(1) processes.The Annals of Statistics, 19(3):1129 –

work page 1991

[9] [9]

Cox, D. D. and Llatas, I. (1991). Maximum likelihood type estimation for nearly nonstationary autoregressive time series.The Annals of Statistics, 19(3):1109 –

work page 1991

[10] [10]

Davis, R. A. (1983). Stable limits for partial sums of dependent random variables.The Annals of Probability, 11(2):262 –

work page 1983

[11] [11]

Dehling, H., Denker, M., and Philipp, W. (1986). Central limit theorems for mixing sequences of random variables under minimal conditions.The Annals of Probability, 14(4):1359 –

work page 1986

[12] [12]

Deuschel, J.-D., Orenshtein, T., and Perkowski, N. (2021). Additive functionals as rough paths. The Annals of Probability, 49(3):1450 –

work page 2021

[13] [13]

Dobrushin, R. L. and Major, P. (1979). Non-central limit theorems for non-linear functionals of Gaussian fields.Zeitschrift f ¨ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 50:27 –

work page 1979

[14] [14]

and Resnick, S

Durrett, R. and Resnick, S. I. (1978). Functional limit theorems for dependent variables.The Annals of Probability, 6(5):829 –

work page 1978

[15] [15]

Dynkin, E. B. (1977). Markov systems and their additive functionals.The Annals of Probability, 5(5):653 –

work page 1977

[16] [16]

Dynkin, E. B. (1991). Branching particle systems and superprocesses.The Annals of Probability, 19(3):1157 –

work page 1991

[17] [17]

29 Getoor, R. K. and Sharpe, M. J. (1973). Last exit times and additive functionals.The Annals of Probability, 1(4):550 –

work page 1973

[18] [18]

Glover, J. (1980). Compactifications for dual processes.The Annals of Probability, 8(6):1119 –

work page 1980

[19] [19]

Hairer, M. (2021). Introduction to Malliavin calculus. Lecture notes, available athttps://www. hairer.org/notes/Malliavin.pdf. Hardy, G. H., Littlewood, J. E., and P´ olya, G. (1988).Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge. Reprint of the 1952 edition. Hong, M., Liu, H., and Xu, F. (2024). Limit theorems for add...

work page 2021

[20] [20]

Jain, N. C. (1990). Large deviation lower bounds for additive functionals of Markov processes. The Annals of Probability, 18(3):1071 –

work page 1990

[21] [21]

Jara, M., Komorowski, T., and Olla, S. (2009). Limit theorems for additive functionals of a Markov chain.The Annals of Applied Probability, 19(6):2270 –

work page 2009

[22] [22]

(2021).Foundations of modern probability, volume 99 ofProbability Theory and Stochastic Modelling

Kallenberg, O. (2021).Foundations of modern probability, volume 99 ofProbability Theory and Stochastic Modelling. Springer, Cham, third edition. Kaspi, H. (1988). Random time changes for processes with random birth and death.The Annals of Probability, 16(2):586 –

work page 2021

[23] [23]

(2006).Introduction to stochastic integration

Kuo, H.-H. (2006).Introduction to stochastic integration. Universitext. Springer, New York. Kuwae, K. (2010). Stochastic calculus over symmetric Markov processes without time reversal. The Annals of Probability, 38(4):1532 –

work page 2006

[24] [24]

Maisonneuve, B. (1975). Exit systems.The Annals of Probability, 3(3):399 –

work page 1975

[25] [25]

(2014).Multiple Wiener-Itˆ o integrals, volume 849 ofLecture Notes in Mathematics

Major, P. (2014).Multiple Wiener-Itˆ o integrals, volume 849 ofLecture Notes in Mathematics. Springer, Cham, second edition. With applications to limit theorems. Marcus, M. B. and Rosen, J. (1996). Gaussian chaos and sample path properties of additive functionals of symmetric Markov processes.The Annals of Probability, 24(3):1130 –

work page 2014

[26] [26]

and Woodroofe, M

Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains.The Annals of Probability, 28(2):713 –

work page 2000

[27] [27]

Nelson, E. (1973). The free Markoff field.Journal of Functional Analysis, 12(2):211 –

work page 1973

[28] [28]

Nevison, C. H. (1976). Potentials of Markov processes without duality.The Annals of Probability, 4(3):497 –

work page 1976

[29] [29]

and de Acosta, A

Ney, P. and de Acosta, A. (1998). Large deviation lower bounds for arbitrary additive functionals of a Markov chain.The Annals of Probability, 26(4):1660 –

work page 1998

[30] [30]

Nourdin, I. (2012). Lectures on Gaussian approximations with Malliavin calculus. 30 Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos.Probability Theory and Related Fields, 145(1):75 –

work page 2012

[31] [31]

Nourdin, I., Peccati, G., and Reinert, G. (2010). Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos.The Annals of Probability, 38(5):1947 –

work page 2010

[32] [32]

and Poly, G

Nourdin, I. and Poly, G. (2013). Convergence in total variation on Wiener chaos.Stochastic Processes and their Applications, 123(2):651–674. Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals.The Annals of Probability, 33(1):177 –

work page 2013

[33] [33]

and Prabhu, N

Pacheco, A. and Prabhu, N. U. (1996). A Markovian storage model.The Annals of Applied Probability, 6(1):76 –

work page 1996

[34] [34]

and Taqqu, M

Peccati, G. and Taqqu, M. S. (2011).Wiener chaos: moments, cumulants and diagrams, volume 1 ofBocconi & Springer Series. Springer, Milan; Bocconi University Press, Milan. A survey with computer implementation, Supplementary material available online. Peccati, G. and Tudor, C. A. (2005). Gaussian limits for vector-valued multiple stochastic integrals. InS´...

work page 2011

[35] [35]

Peligrad, M., Utev, S., and Wu, W. B. (2007). A maximalL p-inequality for stationary sequences and its applications.Proceedings of the American Mathematical Society, 135(2):541 –

work page 2007

[36] [36]

and Taqqu, M

Pipiras, V. and Taqqu, M. S. (2010). Regularization and integral representations of Hermite processes.Statistics & Probability Letters, 80(23–24):2014 –

work page 2010

[37] [37]

Sabzikar, F. (2015). Tempered Hermite process.Modern Stochastics: Theory and Applications, 2(4):327 –

work page 2015

[38] [38]

Tsybakov, A. B. (2009).Introduction to nonparametric estimation. Springer Series in Statis- tics. Springer, New York. Revised and extended from the 2004 French original, Translated by Vladimir Zaiats. 31 Appendix A Discrete-time stationary Gauss-Markov process The following lemma is well-known, yet we did not find an appropriate discrete-time refer- ence ...

work page 2009

[39] [39]

Ifα= 1, the process is constant in time whereasα=−1 contradicts stationarity

= Var(X 0). Ifα= 1, the process is constant in time whereasα=−1 contradicts stationarity. Thus, it remains to study|α|<1. Proposi- tion 14.7 in Kallenberg (2021) reveals Cov(X0, Xk+1)Var(X1) = Cov(X0, X1)Cov(X1, Xk+1). By stationarity, Var(X1) = Var(X0) and Cov(X1, Xk+1) = Cov(X0, Xk), hence Cov(X0, Xk+1) = Cov(X0, X1) Var(X0) Cov(X0, Xk) =αCov(X 0, Xk).(...

work page 2021

[40] [40]

As concernsl < k, we writeε l =X l −µ−α(X l−1 −µ)

= Var(X0) +α 2Var(X0)−2α 2Var(X0) = Var(X0)(1−α 2). As concernsl < k, we writeε l =X l −µ−α(X l−1 −µ). Thenε l is represented as a function ofX l andX l−1, both in the past of time-stepk. Due to previous observations, εk ⊥ ⊥εl and more generallyε k ⊥ ⊥(ε1, . . . , εk−1). By induction, the sequence (ε k)k∈N is i.i.d. Finally, isolatingX k in the defining e...

work page 2009