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arxiv: 2604.17705 · v1 · submitted 2026-04-20 · 🧮 math.ST · stat.TH

Asymptotic behavior of the variance of the BLUE for the mean of stationary processes

Mamikon S. Ginovyan This is my paper

Pith reviewed 2026-05-10 04:09 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords BLUE varianceasymptotic behaviorstationary processesspectral densitymean estimationnondeterministic modelsdeterministic modelshyperbolic decay
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The pith

The asymptotic variance of the BLUE for the mean of a stationary process depends only on the spectrum near the origin.

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

This survey establishes that the rate at which the variance of the best linear unbiased estimator for the mean goes to zero in stationary processes is fixed entirely by the spectral density's behavior close to zero frequency. Nondeterministic models produce power-law or hyperbolic decay in that variance, while purely deterministic models permit exponential decay. A necessary condition for the exponential rate is that the spectral density must be zero on a set of positive Lebesgue measure inside every neighborhood of zero. The work also records when other linear unbiased estimators achieve the same asymptotic efficiency as the BLUE. These facts matter because they isolate the low-frequency part of the spectrum as the sole driver of long-run estimation precision for the mean in time series.

Core claim

The asymptotic behavior of the variance of the BLUE for the mean of stationary processes is determined solely by the behavior of the spectrum near the origin. For nondeterministic models the variance exhibits hyperbolic behavior similar to a power function, while for purely deterministic models the variance decreases at an exponential rate. A necessary condition for the variance to approach zero exponentially is that the spectral density of the model vanishes on a set of positive Lebesgue measure in any neighborhood of zero. The survey further records the asymptotic efficiency of various other unbiased linear estimators relative to the BLUE.

What carries the argument

The spectral density's local behavior near the origin, which fixes the decay rate of the BLUE variance through the regularity and memory structure of the stationary process.

If this is right

  • Processes with positive spectral density at zero have BLUE variance that decays hyperbolically.
  • Purely deterministic stationary processes allow the BLUE variance to decay exponentially fast.
  • Other linear unbiased estimators match the BLUE asymptotically whenever the spectrum near zero governs the rate.
  • Exponential decay requires the spectral density to vanish on a positive Lebesgue measure set in every neighborhood of zero.

Where Pith is reading between the lines

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

  • Knowledge of the spectrum only near zero would suffice to predict the long-run precision of the sample mean without specifying the full process.
  • The efficiency comparisons imply that computationally simpler linear estimators often lose nothing asymptotically when the low-frequency spectrum is the controlling factor.
  • These rates suggest that accurate estimation of the spectrum at the lowest frequencies is the practical bottleneck for reliable mean inference in long stationary records.

Load-bearing premise

The stationary processes have regularity and memory structures such that the spectral density near the origin alone fixes the asymptotic variance of the BLUE.

What would settle it

A concrete stationary process whose spectral density is positive at zero yet whose BLUE variance decays exponentially, or whose spectral density vanishes on a positive-measure set near zero yet whose BLUE variance decays only hyperbolically, would refute the stated necessary condition and the sole-determination claim.

read the original abstract

In this paper, we survey results on the asymptotic behavior of the variance of the best linear unbiased estimator (BLUE) for the mean of stationary processes. This behavior is influenced by the regularity and memory structures of the observed models. The results show that the asymptotic behavior of the variance of the BLUE is determined solely by the behavior of the spectrum near the origin. For nondeterministic models, the variance of the BLUE exhibits hyperbolic behavior, similar to the power function, while for purely deterministic models, the variance decreases at an exponential rate. Specifically, a necessary condition for the variance of the BLUE to approach zero exponentially is that the spectral density of the model vanishes on a set of positive Lebesgue measure in any neighborhood of zero. We also present results on the asymptotic efficiency of various unbiased linear estimators in comparison to the BLUE.

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 / 3 minor

Summary. The manuscript is an expository survey of classical results on the asymptotic variance of the best linear unbiased estimator (BLUE) for the mean of a stationary process X_t = μ + Y_t. It asserts that this asymptotic rate is governed exclusively by the low-frequency behavior of the spectral measure: hyperbolic (power-law) decay when the spectrum is absolutely continuous near zero (nondeterministic case), and exponential decay when the spectral measure is singular with a gap at the origin (purely deterministic case). A necessary condition for the exponential rate is that the spectral density vanishes on a set of positive Lebesgue measure in every neighborhood of zero. The paper also compares the asymptotic efficiency of other unbiased linear estimators to the BLUE.

Significance. If the survey accurately recapitulates the standard spectral representations of the quadratic form 1^T Σ_n^{-1} 1 and the associated Toeplitz inverse properties, the consolidated statement of these classical limits is a useful reference for time-series practitioners working with long-memory or deterministic components. The reduction to low-frequency behavior is a standard consequence of the spectral theorem for stationary processes and does not introduce new technical machinery.

minor comments (3)
  1. §2 (or equivalent introductory section on notation): the distinction between 'nondeterministic' and 'purely deterministic' models is introduced via spectral properties but would benefit from an explicit reference to the Wold decomposition or the Lebesgue decomposition of the spectral measure to avoid ambiguity for readers unfamiliar with the classical terminology.
  2. The statement of the necessary condition for exponential decay (spectral density vanishes on positive-measure set near zero) is correct but would be strengthened by citing the precise theorem (e.g., the relevant result on the decay of the inverse Toeplitz quadratic form) rather than leaving it as a survey paraphrase.
  3. Figure or table comparing rates across examples (if present) should include the exact spectral densities used, so that the hyperbolic vs. exponential distinction can be verified numerically by the reader.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately reflects the scope of our expository survey on the asymptotic variance of the BLUE for the mean of stationary processes, including the key role of low-frequency spectral behavior and the distinction between hyperbolic and exponential decay rates.

Circularity Check

0 steps flagged

No significant circularity; expository survey of classical results

full rationale

This paper is an expository survey of classical results on the asymptotic behavior of the variance of the BLUE for the mean of stationary processes. It references external results without presenting original derivations that could introduce circularity. The central claims rest on standard spectral representations and properties of Toeplitz matrices, which are well-established in the literature and not derived internally in a self-referential manner. No load-bearing steps reduce to self-definition, fitted inputs renamed as predictions, or self-citation chains within the paper itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The survey relies on standard mathematical assumptions in the theory of stationary processes, such as the existence of a spectral density and stationarity, without introducing new free parameters or entities.

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Works this paper leans on

63 extracted references · 63 canonical work pages

  1. [1]

    Adenstedt, R.K. (1974). On large-sample estimation for the mean of a stationary random sequence. Ann. Math. Statist 2, 1095--1107

    work page 1974

  2. [2]

    and Eisenberg B

    Adenstedt, R.K. and Eisenberg B. (1974). Linear estimation of regression coefficients. Quart. Appl. Math. 32 (3), 317--327

    work page 1974

  3. [3]

    Babayan, N. M. and Ginovyan, M. S. (2019). Asymptotic behavior of the variance of the best linear unbiased estimator for the mean of a discrete-time singular stationary process J. Cont. Math. Anal. 54 (6), 319--328

    work page 2019

  4. [4]

    Babayan, N. M. and Ginovyan, M. S. (2019). On exponential decay of the variance of BLUE for the mean of a stationary sequence. Studia Sci. Math. Hungarica 56 (4), 482--491

    work page 2019

  5. [5]

    Babayan, N. M. and Ginovyan, M. S. (2023). Asymptotic behavior of the prediction error for stationary sequences. Probability Surveys 20, 664--721

    work page 2023

  6. [6]

    Babayan, N. M. , Ginovyan, M. S. and Taqqu, M.S. (2021). Extensions of Rosenblatt's results on the asymptotic behavior of the prediction error for deterministic stationary sequences. J. Time Ser. Anal. 42, 622--652

    work page 2021

  7. [7]

    New directions in time series analysis, part II

    Beran, J. (1993). Recent developments in location estimation and regression for long-memory processes. In "New directions in time series analysis, part II" (D. Brillinger et al. eds.), vol. Volume 46, 1-10, Springer-Verlag

    work page 1993

  8. [8]

    and Kulik, R

    Beran, J., Feng, Y., Ghosh, S. and Kulik, R. (2013). Long-Memory Processes: Probabilistic Properties and Statistical Methods. Springer-Verlag, Berlin

    work page 2013

  9. [9]

    unsch, H. (1985) Location estimates for processes with long range dependence. Research Report No. 40, Seminar f\

    Beran, J. and K\"unsch, H. (1985) Location estimates for processes with long range dependence. Research Report No. 40, Seminar f\"ur Statistik, Eiden\"ossische Technische Hochschule, Z\"urich

    work page 1985

  10. [10]

    , and Teugels, J.L

    Bingham, N.H., Goldie, C.M. , and Teugels, J.L. (1989). Regular Variation. Cambridge University Press, Cambridge

    work page 1989

  11. [11]

    Bleher, P. M. (1981). Inversion of Toeplitz matrices. Trans. Moscow Math. Soc. 2, 201--229

    work page 1981

  12. [12]

    (1991) Time Series: Theory and Methods, 2nd ed

    Brockwell, P.J and Davis, R.A. (1991) Time Series: Theory and Methods, 2nd ed. Springer-Verlag, New York

    work page 1991

  13. [13]

    and Nessel, R.J

    Butzer, P.L. and Nessel, R.J. (1971). Fourier Analysis and Approximation I. Academic Press, New York

    work page 1971

  14. [14]

    and Leadbetter, M.R

    Cram\'er, H. and Leadbetter, M.R. (1967). Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications. Wiley, New York

    work page 1967

  15. [15]

    Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17, 1749--1766

    work page 1989

  16. [16]

    Dahlhaus, R. (1990). Approximations for the inverse of Toeplitz matrices with applications to stationary processes. Linear Algebra Appl. 127, 27--40

    work page 1990

  17. [17]

    Doob, J.L. (1953). Stochastic Processes, Wiley, New York

    work page 1953

  18. [18]

    Erd\'elyi, A. et al. (1953). Higher Transcendental Functions, Vol. 1 and 2. McGraw-Hill, New York

    work page 1953

  19. [19]

    Fekete, M. (1930). \"Uber den transfiniten Durchmesser ebener Punktmengen. Zweite Mitteilung. Math. Z. 32, 215--221

    work page 1930

  20. [20]

    Geronimus, Ya. L. (1965). Some asymptotic properties of orthogonal polynomials, Soviet Math. Dokl. 6 1387--1389. Russian original: Dokl. Akad. Nauk SSSR bf 165 (1965), 19--20

    work page 1965

  21. [21]

    and Skorokhod, A.V

    Gihman, I.I. and Skorokhod, A.V. (1980). The Theory of Stochastic Processes I. Springer, New York

    work page 1980

  22. [22]

    Ginovyan, M. S. (2025). Random Toeplitz Functionals and Their Applications. Springer

    work page 2025

  23. [23]

    Golinskii, B. L. (1958). Certain asymptotic relations in the theory of orthogonal polynomials (Russian). Izv. Vysh. Uchebn. Zaved. Matematika 2(3), 29--38

    work page 1958

  24. [24]

    Golinskii, B. L. (1966). Certain estimates for the Christoffel-Darboux kernels and for moduli of orthogonal polynomials (Russian). Izv. Vysh. Uchebn. Zaved. Matematika 50(1), 30--42

    work page 1966

  25. [25]

    Golinskii, B. L. (1974). The V. A. Steklov problem in the theory of orthogonal polynomials Math. Notes 15(1), 13--19

    work page 1974

  26. [26]

    Goluzin, G. M. (1969). Geometric Theory of Functions of a Complex Variable. Amer. Math. Soc., Providence

    work page 1969

  27. [27]

    (1950) Stochastic processes and statistical inference, Arkiv f\"or Matematik, 1, 17, 195-277

    Grenander, U. (1950) Stochastic processes and statistical inference, Arkiv f\"or Matematik, 1, 17, 195-277

    work page 1950

  28. [28]

    Grenander, U. (1952). On Toeplitz forms and stationary processes. Ark. Mat. 1 (37), 555--571

    work page 1952

  29. [29]

    Grenander, U. (1954). On the estimation of regression coefficients in the case of an autocorrelated disturbance. Ann. Math. Statist. 25, 252-272

    work page 1954

  30. [30]

    Grenander, U. (1981). Abstract Inference. Wiley, New York

    work page 1981

  31. [31]

    and Rosenblatt, M

    Grenander, U. and Rosenblatt, M. (1954). An Extension of a Theorem of G. Szeg o and its Application to the Study of Stochastic Processes. Trans. Amer. Math. Soc. 76, 112--126

    work page 1954

  32. [32]

    and Rosenblatt, M

    Grenander, U. and Rosenblatt, M. (1957). Statistical Analysis oj Stationary Time Series. Wiley, New York

    work page 1957

  33. [33]

    and Szeg o , G

    Grenander, U. and Szeg o , G. (1958). Toeplitz Forms and Their Applications. University of California Press, Berkeley and Los Angeles

    work page 1958

  34. [34]

    Hoffman, K. . (1958). Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliffs, N.J

    work page 1958

  35. [35]

    and Linnik, Yu

    Ibragimov, I.A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing Groningen, The Netherlands

    work page 1971

  36. [36]

    Ibragimov, I. A. and Rozanov. Yu.A. (1978). Gaussian Random Processes. Springer, New York

    work page 1978

  37. [37]

    Kholevo, A.S. (1971). On the asymptotic efficiency of pseudobest estimates. Theory Probab. Appl. 16(3), 516--527

    work page 1971

  38. [38]

    Kolmogorov, A. N. (1941). Stationary sequences in a Hilbert space. Bull. Moscow State University 2(6), 1--40

    work page 1941

  39. [39]

    Kolmogorov, A. N. (1941). Interpolation and Extrapolation of stationary random sequences. Izv. Akad. Nauk SSSR. Ser. Mat. 5, 3--14

    work page 1941

  40. [40]

    Lo\'eve, M. (1978). Probability Theory II. Springer, New York

    work page 1978

  41. [41]

    and Totik, V

    M\'at\'e, A., Nevai, P. and Totik, V. (1991). Szeg o 's extremum problem on the unit circle. Ann. of Math., 134, 433--453

    work page 1991

  42. [42]

    Mazurkievicz, S. (1945). Un theoreme sur les polynomes. Ann. Soc. Polon. Math.\/ 18 , 113--118

    work page 1945

  43. [43]

    Nevai, P. (1986). Geza Freud, Orthogonal Polynomials and Christoffel Functions: A Case Study. J. Approx. Theory, 48 , 3--167

    work page 1986

  44. [44]

    M and Sorokin, V

    Nikishin, E. M and Sorokin, V. N. (1991). Rational approximations and orthogonality.\/ Translations of Mathematical Monographs, v. 92, American Mathematical Society, Providence, Rhode Island

    work page 1991

  45. [45]

    (1961) An approach to time series analysis

    Parzen, E. (1961) An approach to time series analysis. Ann. Math. Statist. 32 (4), 951--989

    work page 1961

  46. [46]

    (1967) Time Series Analysis Papers, Holden-Day, San Francisco

    Parzen, E. (1967) Time Series Analysis Papers, Holden-Day, San Francisco

    work page 1967

  47. [47]

    Pourahmadi, M. (2001). Fundamentals of Time Series Analysis and Prediction Theory. Wiley, New York

    work page 2001

  48. [48]

    and Seghier, A

    Rambour, P. and Seghier, A. (2009). Asymptotic inversion of Toeplitz matrices with one singularity in the symbol. C. R. Acad. Sci. Paris, Ser. I, 347, 489--494

    work page 2009

  49. [49]

    Rasulov, N.P. (1976). On asymptotically efficient estimates of regression coefficients under spectral density of noise degeneration. Theory Probab. Appl. 21(2), 316--324

    work page 1976

  50. [50]

    and Kholevo, A.S

    Rasulov, N.P. and Kholevo, A.S. (1978). A regression problem for time-continuous processes. Theory Probab. Appl. 23(4), 731--740

    work page 1978

  51. [51]

    Rosenblatt, M. (1957). Some Purely Deterministic Processes. J. Math. and Mech. 6(6), 801--810. (Reprinted in: Selected works of Murray Rosenblatt, Davis R.A, Lii K.-S., Politis D.N. eds., Springer, New York, 124--133, (2011)

    work page 1957

  52. [52]

    and Taqqu, M

    Samarov, A. and Taqqu, M. S. (1988). On the efficiency of the sample mean in long-memory noise. J. Time Series Analysis , 9 , 191--200

    work page 1988

  53. [53]

    Simon, B. (2005). Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory, Part 2: Spectral Theory. AMS Colloqium Publications, Vol. 54, Parts 1,2, Providence, Rhode Island

    work page 2005

  54. [54]

    Simon, B. (2011). Szeg o ’s Theorem and Its Descendants: Spectral Theory for L^2 Perturbations of Orthogonal Polynomials . Princeton University Press, Princeton and Oxford

    work page 2011

  55. [55]

    Simon, B. (2015). Real Analysis. A Comprehensive Course in Analysis, Part 1 . AMS, Providence, Rhode Island

    work page 2015

  56. [56]

    Szeg o , G. (1915). Ein Grenzwertsatz \"uber die Toeplitzschen Determinanten einer reellen positiven Funktion. Math. Ann. 76, 490--503

    work page 1915

  57. [57]

    Uber orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene geh\

    Szeg o , G. (1921). \"Uber orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene geh\"oren. Math. Zeitschrift. 9, 218--270

    work page 1921

  58. [58]

    Szeg o , G. (1939). Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ., 23, American Mathematical Society, Providence (3rd edition, 1967)

    work page 1939

  59. [59]

    Szeg o , G. (1950). On certain special sets of orthogonal polynomials. Proc. Amer. Math. Soc. 1, 731--737

    work page 1950

  60. [60]

    Szeg o , G. (1982). Collected Papers, Vol. I, edited by Richard Askey, Birkh\"auser, Boston-Basel-Stuttgart

    work page 1982

  61. [61]

    Vitale, R.A. (1973). An asymptotically efficient estimate in time series analysis. Quart. Appl. Math. 30 , 421--440

    work page 1973

  62. [62]

    Yajima, Y. (1988). On estimation of a regression model with long-memory stationary errors, Ann. Statist. 16, 791-807

    work page 1988

  63. [63]

    Zygmund, A. ( 1959). Trigonometric Series, vol. 1, 2nd ed., University Press, Cambridge

    work page 1959