pith. sign in

arxiv: 1907.01757 · v1 · pith:3EE3XPI4new · submitted 2019-07-03 · 🧮 math.PR · math.DS

Cram\'{e}r type moderate deviations for stationary sequences of bounded random variables

Xiequan Fan This is my paper

Pith reviewed 2026-05-25 10:22 UTC · model grok-4.3

classification 🧮 math.PR math.DS
keywords moderate deviationsstationary sequencesbounded random variablesCramér typemoderate deviation principleBerry-Esseen boundφ-mixing sequencesMarkov chains
0
0 comments X

The pith

Stationary sequences of bounded random variables satisfy Cramér type moderate deviations.

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

The paper derives exponential inequalities of Cramér type that control moderate deviations for partial sums of stationary bounded random variables. These inequalities sit between the central limit theorem regime and full large deviation principles. A sympathetic reader would care because the bounds deliver explicit tail estimates that also produce a moderate deviation principle and a Berry-Esseen bound. The same machinery is applied to quantile coupling, functions of φ-mixing sequences, and contracting Markov chains.

Core claim

We derive Cramér type moderate deviations for stationary sequences of bounded random variables. Our results imply the moderate deviation principles and a Berry-Esseen bound. Applications to quantile coupling inequalities, functions of φ-mixing sequences, and contracting Markov chains are discussed.

What carries the argument

Cramér type moderate deviation inequalities for the normalized partial sums under the stationarity and boundedness assumptions.

If this is right

  • The moderate deviation principle holds for the partial sums.
  • A Berry-Esseen bound follows for the normal approximation error.
  • Quantile coupling inequalities are obtained as a corollary.
  • The results apply directly to functions of φ-mixing sequences.
  • Contracting Markov chains satisfy the same moderate deviation bounds.

Where Pith is reading between the lines

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

  • The stationarity assumption might be weakened to ergodicity or other weak dependence conditions.
  • Similar bounds could be checked numerically on simple AR(1) processes with bounded innovations.
  • The derived Berry-Esseen rate might be compared against existing rates for mixing sequences.

Load-bearing premise

The random variables are bounded and the sequence is stationary.

What would settle it

A stationary sequence of bounded random variables for which the moderate deviation probabilities fail to obey the stated Cramér-type exponential bounds.

read the original abstract

We derive Cram\'{e}r type moderate deviations for stationary sequences of bounded random variables. Our results imply the moderate deviation principles and a Berry-Esseen bound. Applications to quantile coupling inequalities, functions of $\phi$-mixing sequences, and contracting Markov chains are discussed.

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

0 major / 1 minor

Summary. The paper derives Cramér-type moderate deviation results for stationary sequences of bounded random variables. These results are shown to imply moderate deviation principles (MDP) as well as a Berry-Esseen bound. Applications are discussed for quantile coupling inequalities, functions of φ-mixing sequences, and contracting Markov chains.

Significance. If the derivations hold under the stated boundedness and stationarity conditions (with appropriate mixing or dependence assumptions), the work would extend moderate deviation theory to a useful class of dependent processes and supply explicit bounds with direct statistical applications. The explicit implication to Berry-Esseen bounds and the listed applications to mixing and Markov settings are concrete strengths.

minor comments (1)
  1. The abstract states the main claims but does not list the precise mixing or dependence conditions under which the Cramér-type moderate deviations hold; a short statement of the minimal assumptions would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript on Cramér-type moderate deviations for stationary sequences of bounded random variables. The assessment correctly identifies the main results (MDP and Berry-Esseen bounds) and the applications to quantile coupling, φ-mixing sequences, and contracting Markov chains. No specific major comments or criticisms are provided in the report, so we have no individual points to rebut or revise at this stage. We remain available to address any concrete questions the referee may wish to raise.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract states the derivation of Cramér-type moderate deviations for stationary bounded sequences, implying MDP and Berry-Esseen bounds, with applications to mixing sequences and Markov chains. No equations, self-citations, fitted parameters, or derivation steps are supplied in the visible text, so no load-bearing reduction to inputs by construction can be exhibited. The skeptic assessment confirms that without the full manuscript, no internal inconsistency or circular step is locatable, and the claims align with standard dependence results under typical mixing assumptions. The derivation is therefore treated as self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is available from the abstract.

pith-pipeline@v0.9.0 · 5555 in / 894 out tokens · 32346 ms · 2026-05-25T10:22:30.760026+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    On martingale approxima tions and the quenched weak inviariance principle

    Cuny, C., Merlev` ede, F., 2014. On martingale approxima tions and the quenched weak inviariance principle. Ann. Probab. 42(2), 760–793

    work page 2014

  2. [2]

    Moderate deviations and associated Laplace approximations for sums of independent random vectors

    De Acosta, A., 1992. Moderate deviations and associated Laplace approximations for sums of independent random vectors. Trans. Amer. Math. Soc. 329(1): 357-375

    work page 1992

  3. [3]

    Moderate deviations for stationary sequences of bounde d random variables

    Dedecker, J., Merlev` ede, F., Peligrad, M., Utev, S., 20 09. Moderate deviations for stationary sequences of bounde d random variables. Ann. Inst. H. Poincar´ e Probab. Statist. 45(2), 453–476

    work page

  4. [4]

    Rates of con vergence for minimal distances in the central limit theorem under projective criteria

    Dedecker, J., Merlev` ede, F., Rio, E., 2009. Rates of con vergence for minimal distances in the central limit theorem under projective criteria. Electron. J. Probab. , 14, 978-1011

    work page 2009

  5. [5]

    Moderate deviations for martingales wi th bounded jumps

    Dembo, A., 1996. Moderate deviations for martingales wi th bounded jumps. Electron. Comm. Proba. 1, 11–17

    work page 1996

  6. [6]

    Moderate deviations for martingale differences and applications to φ−mixing sequences

    Djellout, H., 2002. Moderate deviations for martingale differences and applications to φ−mixing sequences. Stochastic Stochastic Rep. 73, 37–63

    work page 2002

  7. [7]

    Cram´ er large deviation expa nsions for martingales under Bernstein’s condition

    Fan X, Grama I, Liu Q., 2013. Cram´ er large deviation expa nsions for martingales under Bernstein’s condition. Stochastic Process. Appl. 123(11): 3919–3942

    work page 2013

  8. [8]

    A., 1975

    Freedman, D. A., 1975. On tail probabilities for marting ales. Ann. Probab. 3(1): 100–118

    work page 1975

  9. [9]

    Q., 1996

    Gao, F. Q., 1996. Moderate deviations for martingales an d mixing random processes. Stochastic Process. Appl. 61, 263–275

    work page 1996

  10. [10]

    On moderate deviations for martingale s

    Grama, I., 1997. On moderate deviations for martingale s. Ann. Probab. 25, 152–184

    work page 1997

  11. [11]

    Large deviations for mar tingales via Cram´ er’s method

    Grama, I., Haeusler, E., 2000. Large deviations for mar tingales via Cram´ er’s method. Stochastic Process. Appl. 85, 279–293

    work page 2000

  12. [12]

    An asymptotic expansion for probabilities of moderate deviations for multivariate martingales

    Grama, I., Haeusler, E., 2006. An asymptotic expansion for probabilities of moderate deviations for multivariate martingales. J. Theoret. Probab. 19, 1–44. 18

    work page 2006

  13. [13]

    Quantile coupling inequ alities and their applications

    Mason, D.A., Zhou, H.H., 2012. Quantile coupling inequ alities and their applications. Probab. Surveys 9: 439–479

    work page 2012

  14. [14]

    A maximal Lp-inequality for stationary sequecens and its applications

    Peligrad, M., Utev, S., W u, W.B., 2007. A maximal Lp-inequality for stationary sequecens and its applications . Proc. Amer. Math. Soc. 135, 541–550

    work page 2007

  15. [15]

    A new maximal inequality a nd invariance principle for stationary sequences

    Peligrad, M., Utev, S., 2005. A new maximal inequality a nd invariance principle for stationary sequences. Ann. Probab. 33: 798–815

    work page 2005

  16. [16]

    On probabilities of large devia tions for martingales

    Raˇ ckauskas, A., 1990. On probabilities of large devia tions for martingales. Liet. Mat. Rink. 30, 784–795

    work page 1990

  17. [17]

    Large deviations for martingal es with some applications

    Raˇ ckauskas, A., 1995. Large deviations for martingal es with some applications. Acta Appl. Math. 38, 109–129

    work page 1995

  18. [18]

    Sur le th´ eoreme de Berry-Esseen pour les suites faiblement d´ ependantes.Probab

    Rio, E., 1996. Sur le th´ eoreme de Berry-Esseen pour les suites faiblement d´ ependantes.Probab. Theory Relat. Fields 104(2), 255–282

    work page 1996

  19. [19]

    Nonlinear system theorey: Another look at dependence

    W u, W.B., 2005. Nonlinear system theorey: Another look at dependence. Proc. Natl. Acad. Sci. USA 102, 14150–14154

    work page 2005

  20. [20]

    Moderate deviations for stati onary processes

    W u, W.B., Zhao, Z., 2008. Moderate deviations for stati onary processes. Statist. Sinica 18, 769–782. 19

    work page 2008