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REVIEW 3 major objections 3 minor 34 references

On the estimation of high-dimensional integrated covariance matrix based on high-frequency data with multiple transactions

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper shows that pre-averaging a time-variation adjusted covariance matrix removes the distorting effects of microstructure noise and multiple transactions per timestamp, so the integrated covariance matrix can be estimated from…

desk verdict New and plausible synchronous result on multiple-transaction TVA matrices; the asynchronous claim in Theorem 2.3 is gated by an unrealistic assumption and should not be sold as eliminating asynchrony. read the letter →

arxiv 1908.08670 v2 pith:ZDXAFMCN submitted 2019-08-23 math.ST stat.APstat.TH

Moming Wang , Ningning Xia , You Zhou This is my paper · ORCID
classification math.STstat.APstat.TH MSC 60B2062H12
keywords integratedcovariancematrixhigh-frequencydatamultipletransactionsmicrostructurenoiselimitingspectraldistributionMarchenko-Pasturequationnonlinearshrinkagehigh-dimensionaldiffusionprocess
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

This paper tries to establish a practical recovery result: when high-frequency prices arrive as several transactions sharing one recorded timestamp, the usual time-variation adjusted realized covariance (TVA) matrix has a limiting eigenvalue distribution that is contaminated by the pattern of those multiple transactions. The authors prove that a pre-averaged version, the PA-ATVA matrix, removes both that contamination and market microstructure noise: its empirical spectral distribution converges almost surely to the distribution determined by the integrated covariance matrix through the Marchenko-Pastur equation. If correct, this means the spectrum of the true integrated covariance, the quantity behind risk and portfolio allocation, can be recovered from noisy, trade-bunched tick data using standard random-matrix inversion algorithms. The paper also proposes three nonlinear shrinkage estimators for the integrated covariance matrix and evaluates them in simulations and on Dow Jones stocks.

What carries the argument

The central object is the time-variation adjusted realized covariance (TVA) matrix, which replaces each return vector by its normalized version $\Delta X_i/|\Delta X_i|$ and rescales by the total realized variation; for class C processes this removes the unknown scalar volatility $\gamma_t$ from the limiting spectral relationship. The paper's new machinery is the pre-averaging averaged TVA (PA-ATVA) matrix $B_M = 3\sum_{i=1}^M |\Delta\tilde Y_{2i}|^2/p \cdot \frac{p}{M}\sum_{i=1}^M \frac{\Delta\tilde Y_{2i}(\Delta\tilde Y_{2i})^T}{|\Delta\tilde Y_{2i}|^2}$, built from returns of prices averaged over blocks of length $h=\lfloor \xi n^\beta\rfloor$ with $\beta\in(1/2,1)$. Averaging over blocks longer than $\sqrt{n}$ damps the microstructure noise, and the factor 3 adjusts the pre-averaging scale; the proof shows the empirical spectrum of $B_M$ has the same almost-sure limit as a sample covariance built from the ICV, so its Stieltjes transform obeys the Marchenko-Pastur equation with aspect ratio $c=\lim p/M$.

What would settle it

Simulate class C prices, additive microstructure noise, and transaction times spread uniformly inside each recording interval so that $\max_{i,q} n h (s_{T_i}^{(q)}-t_i)$ does not converge to $0$. Compute the PA-ATVA matrix $B_M$ and compare its empirical spectral distribution with the Marchenko-Pastur transform of the true ICV spectrum for growing $p$ and $n$. If the maximum distance between these distributions does not shrink, the paper's asynchronous-elimination claim fails; under the paper's own simulation protocol, in which the condition holds, the distance does shrink.

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Extended reading notes

Core claim

Working with class C diffusion processes, for which the covolatility matrix factorizes as $\Theta_t = \gamma_t \Lambda$, the paper proves that in the high-dimensional setting multiple transactions break the classical TVA limit. Without noise, the averaged TVA matrix has a limiting spectral distribution that depends on the distribution of the numbers of transactions per timestamp as well as on the ICV spectrum; the paper constructs an adjusted version whose limit depends on the ICV alone when the transaction-count process is piecewise constant. With microstructure noise, the pre-averaging averaged TVA matrix $B_M$ has an empirical spectral distribution that converges almost surely to a law $F_B$ whose Stieltjes transform satisfies the Marchenko-Pastur equation $m_B(z)=\int \frac{1}{\tau(1-c(1+z m_B(z)))-z}\,dH(\tau)$, where $H$ is the limiting spectral distribution of the integrated covariance matrix. Hence pre-averaging removes both microstructure noise and the multiple-transaction effect, and the spectrum of the ICV can be recovered by standard random-matrix inversion. An asynchronous generalization is claimed when, for every stock and every recording interval, the last transaction time is within $o(1/(nh))$ of the recorded timestamp.

Load-bearing premise

The load-bearing assumption is that every stock's last transaction in each recording interval lands within $o(1/(nh))$ of the recorded timestamp; if trades are spread through the interval, the discarded residual terms need not vanish.

Editorial extensions

If this is right

  • Standard random-matrix inversion algorithms can estimate the eigenvalues of the integrated covariance matrix from the PA-ATVA matrix, since both share the same Marchenko-Pastur relation.
  • The limiting distribution of transaction counts no longer matters after pre-averaging, so one estimator works when stocks have very different numbers of trades per timestamp.
  • Using the plain TVA matrix without pre-averaging becomes unreliable once multiple transactions are common, because its spectrum depends on the unknown transaction-count process.
  • The proposed NS, ANS, and MNS estimators give concrete, computable ICV estimates; MNS has the best relative Frobenius loss in spiked or factor-changing simulations, while ANS attains the lowest portfolio risk in the empirical study.

Reading between the lines

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

  • I would expect the asynchronous version of the result to fail on real data where hundreds of trades occur throughout a timestamp, because the key condition demands the last trade occur within $o(1/(nh))$ of the stamp; a coarser timestamp or quote-based observation may be needed to fulfill it.
  • The same pre-averaging mechanism should yield a simple one-dimensional volatility estimator that is immune to the transaction count process, by applying the factor-3 correction to pre-averaged squared returns; the paper does not isolate this scalar case.
  • Since the theorem connects the ICV spectrum to $B_M$ through the Marchenko-Pastur equation, the residual distance between empirical and predicted spectra could be turned into a specification test for the class C assumption; the paper does not propose such a test.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. Using random matrix theory, this paper studies estimation of integrated covariance (ICV) matrices when high-frequency data contain multiple transactions per recording timestamp. In the noiseless case, Theorem 2.1 characterizes the LSD of an averaged TVA matrix as depending on the second moment of the transaction counts, and Corollary 2.1 introduces an adjusted A-ATVA matrix whose LSD is linked to that of ICV through the Marchenko-Pastur equation. For noisy data, Theorem 2.2 states that a pre-averaged PA-ATVA matrix has an LSD determined solely by the ICV spectrum, and Theorem 2.3 claims an extension to asynchronous transaction times. Three nonlinear shrinkage estimators (NS, ANS, MNS) are proposed; simulations and a DJIA minimum-variance portfolio application illustrate the methods.

Significance. Conditional on the synchronous results, this is a useful extension of Zheng and Li (2011) and Xia and Zheng (2018) to multiple transactions. The proof structure of Theorems 2.1 and 2.2 is plausible, the supplementary material is detailed, and the simulation section directly targets the Marchenko-Pastur relationship, which gives the main result concrete support. The A-ATVA correction for the noiseless transaction-count effect is a nice contribution. However, the asynchronous-trading claim is not established: Theorem 2.3 requires transaction end times within o(1/(nh)) of the recording timestamps, and the paper's own Table 2 shows that stock-specific transaction counts break the corresponding noiseless relation. These issues are load-bearing for the advertised elimination of asynchrony, so the paper needs substantial revision before the claims can be accepted as stated.

major comments (3)
  1. [Theorem 2.3 and Appendix C] The asynchronous extension is gated by the condition max_{i,q} n h (s_Ti^(q) - t_i) -> 0. Since s_Ti^(q) <= t_i and h = n^beta with beta > 1/2, this requires the last transaction in every recording interval to fall within o(n^{-(1+beta)}) of the recorded timestamp, whereas the interval itself has length 1/n. Table 1 and Figure 1 show transactions spread throughout the interval, with hundreds of transactions per second, so the condition is not a realistic description of the data used in Section 4.4. In the proof, the residual R_i3 has Var = o(1/n) only under this condition; if the last-transaction gap is O(1/n), the Appendix C calculation gives Var(R_i3^(q)) = O(h/n), so sqrt(p) R_i3 is O_p(1) rather than o(1), and Lemma D.1's o(1/sqrt(p)) perturbation threshold is exactly violated. Thus Theorem 2.3 proves a synchronous-endpoint result rather than elimination of asynchronous trading. The abstract, Section 2.2, and Section 5 should either be restricted to the proved setting or the theorem should be extended under transaction-time assumptions that allow spreads within intervals.
  2. [Section 4.1, Table 2; Section 5] The paper's own simulation shows the load-bearing nature of the asynchrony issue in the noiseless A-ATVA: when L_i^(q) differs across stocks, the maximum distance between the ESDs of the A-ATVA matrix and the sample covariance matrix remains around 0.13 as n increases (Table 2), and the text states that the limiting spectral distribution in that case remains an open problem. The Conclusion's sentence claiming that the proposed approach 'eliminated the effects of microstructure noise and asynchronous trading within one recording time stamp' is therefore broader than what has been proved. This overclaim should be removed or explicitly qualified in the abstract and conclusion.
  3. [Section 4.2, Design II] Design II, which is used to support Theorem 2.3, does not report the positioning of transaction times within each recording interval. If the transaction times are equally spaced within each interval, then s_Ti^(q) - t_i is of order -1/(n L_i^(q)), and the condition of Theorem 2.3 fails for h = 252 and L_i^(q) around 5; if instead all transactions are placed exactly at t_i, the design is not asynchronous. The matching ESDs in Figure 7 therefore cannot be attributed to Theorem 2.3 without specifying how the transaction times were generated. Please state the transaction-time model used in Designs I and II and verify whether Theorem 2.3's assumption holds there.
minor comments (3)
  1. [Section 4.2, Assumption (B.ii)] The simulations use pre-averaging window h = floor(n^0.55), but Assumption (B.ii) requires beta > (3 + ell)/(2 ell + 2). This holds for beta = 0.55 only if ell is sufficiently large, which is plausible for iid noise but should be stated explicitly.
  2. [Equation (4.5)] The displayed formula for the annualized standard deviation has unbalanced parentheses; it should presumably be (w_i^T r_i - hat_mu/(251 - ell))^2 inside the summation.
  3. [Table 4] The column headers 'Averged obs.' and 'Efficient 10-seconds' contain typographical errors; the intended terms are 'Averaged obs.' and 'Effective 10-seconds'.

Circularity Check

0 steps flagged · score 0.0 of 10

No meaningful circularity: the limiting spectral results are derived against external random-matrix benchmarks, and the Xia–Zheng self-citation is independent support rather than a circular load.

full rationale

The paper's limiting spectral results are derived, not assumed: Theorem 2.1 is proved via Proposition A.1 (a Silverstein-type deterministic equivalence for ~Σ) and Proposition A.2 (an explicit law-of-large-numbers computation of the scaling factor, Eq. A.7), with all steps in the supplement. Corollary 2.1 removes the L-dependent nuisance factor by a construction that follows algebraically from Eq. (A.7); no parameter is fitted to the target spectral distribution. Theorem 2.2 proves the two ingredients — noise negligibility (B.3)-(B.4) and consistency 3Σ|Δ~Y|²/p→θ (B.2) — and then invokes the published Xia-Zheng (2018) synchronous pre-averaging result; that citation is a parameter-free external theorem whose assumptions do not include the multiple-transaction or asynchrony claims made here, so it is independent support rather than a circular self-citation. Theorem 2.3 similarly proves residual bounds (C.3)-(C.4) and reduces to the same synchronous theorem. The only notable concern is non-circular: Theorem 2.3's hypothesis max_{i,q} nh(s_Ti−t_i)→0 is much stronger than real-world asynchrony, so the conclusion's phrase 'eliminates ... asynchronous trading' overstates the theorem's scope; but that is an assumption/correctness issue, not a reduction of the result to its inputs. No equation is defined in terms of the target LSD, and no fitted constant is relabeled as a prediction.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

No constants are fitted to reproduce the ICV spectrum; the theoretical claims are derivations. The main nonstandard inputs are the class C structure and the treatment of transaction counts as exogenous, bounded, and, in the asynchronous theorem, nearly simultaneous with the recording stamp. These are domain assumptions rather than fitted parameters, and no new physical or statistical entities are introduced.

free parameters (3)
  • pre-averaging window length h = h = floor(n^0.55) = 252 in simulations; 15 minutes in the empirical study
    Tuning parameter of the PA-ATVA estimator. The theory permits a range of h values, so it is not fitted to the ICV spectrum.
  • MNS window length k_n = k_n = floor(0.75 n^1/2) = 114 in simulations; six minutes in the empirical study
    Hand-chosen window for the integrated volatility estimator along eigenvectors in the mixed nonlinear shrinkage estimator.
  • ANS first-split size M_tau1 = Chosen from candidate set G in (3.3) by minimizing the Frobenius criterion (3.2)
    Data-driven tuning parameter for the averaged nonlinear shrinkage estimator; not fitted to the population ICV.
assumptions (8)
  • standard math Silverstein's theorem on the LSD of sample covariance matrices (Silverstein 1995).
    Used as the external benchmark connecting the limit of a sample covariance built from i.i.d. draws from the ICV matrix to the ICV spectrum.
  • standard math Marchenko-Pastur equation and Stieltjes transform inversion.
    Fundamental random matrix tool used throughout to relate the LSD of an estimator matrix to that of the population matrix.
  • domain assumption Class C factorization Theta_t = gamma_t Lambda with deterministic Lambda Lambda^T.
    Definition 1.1; all theorems restrict the covolatility process to this one-factor-like structure, which is a substantive modeling assumption.
  • domain assumption Transaction counts {L_i} are independent of the price process and have stable means such as E(1/L_i^2) = f2_ti.
    Assumption (A.x); exogeneity of observation times is needed for the averaging arguments and is questionable in real markets where trading intensity correlates with volatility.
  • domain assumption Transaction counts are uniformly bounded by a constant L*.
    Assumption (B.iii); controls the mixing properties of averaged microstructure noise and prevents L_i from growing with n.
  • domain assumption Asynchronous timing condition max_{i,q} n h (s_Ti^(q) - t_i) -> 0.
    Theorem 2.3 premise; the most fragile condition for the claimed elimination of asynchronous trading effects.
  • domain assumption Pre-averaging window h = floor(xi n^beta) with beta in an admissible asymptotic range.
    Assumption (B.ii); required to make the microstructure noise contribution vanish in the limit.
  • domain assumption The function f2* is piecewise constant on a known partition of [0,1].
    Corollary 2.1 assumes a user-specified partition on which E(1/L_i^2) is constant; this partition is not known in practice and is an additional input to the A-ATVA estimator.

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Pith. "Pith review of On the estimation of high-dimensional integrated covariance matrix based on high-frequency data with multiple transactions." pith.science (2026). https://pith.science/paper/ZDXAFMCN

@misc{pith2026190808670,
  author       = {Pith},
  title        = {Pith review of: On the estimation of high-dimensional integrated covariance matrix based on high-frequency data with multiple transactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZDXAFMCN}},
  note         = {Machine review of arXiv:1908.08670}
}
read the original abstract

Due to the mechanism of recording, the presence of multiple transactions at each recording time becomes a common feature for high-frequency data in financial market. Using random matrix theory, this paper considers the estimation of integrated covariance (ICV) matrices of high-dimensional diffusion processes based on multiple high-frequency observations. We start by studying the estimator, the time-variation adjusted realized covariance (TVA) matrix, proposed in Zheng and Li (2011) without microstructure noise. We show that in the high-dimensional case, for a class C of diffusion processes, the limiting spectral distribution (LSD) of averaged TVA depends not only on that of ICV, but also on the numbers of multiple transactions at each recording time. However, in practice, the observed prices are always contaminated by the market microstructure noise. Thus the limiting behavior of pre-averaging averaged TVA matrices is studied based on the noisy multiple observations. We show that for processes in class C, the pre-averaging averaged TVA has desirable properties that it eliminates the effects of microstructure noise and multiple transactions, and its LSD depends solely on that of the ICV matrix. Further, three types of nonlinear shrinkage estimators of ICV are proposed based on high-frequency noisy multiple observations. Simulation studies support our theoretical results and show the finite sample performance of the proposed estimators. At last, the high-frequency portfolio strategies are evaluated under these estimators in real data analysis.

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Works this paper leans on

34 extracted references · 32 canonical work pages

  1. [1]

    A\" i t-Sahalia, Y., Fan, J., and Xiu, D. (2010). High-frequency covariance estimates with noisy and asynchronous financial data. Journal of the American Statistical Association , 105(492):1504--1517

    work page 2010

  2. [2]

    A\" i t-Sahalia, Y., Mykland, P., and Zhang, L. (2005). How often to sample a continuous-time process in the presence of market microstructure noise. Review of Financial Studies , 18(2):351--416

    work page 2005

  3. [3]

    A\" i T-Sahalia, Y., Mykland, P., and Zhang, L. (2011). Ultra high frequency volatility estimation with dependent microstructure noise. Journal of Econometrics , 160(1):160--175

    work page 2011

  4. [4]

    and Bollerslev, T

    Andersen, T. and Bollerslev, T. (1998). Deutsche mark-dollar volatility: intraday activity patterns, macroeconomic announcements, and longer run dependencies. Journal of Finance , 53(1):219--265

    work page 1998

  5. [5]

    Andersen, T., Bollerslev, T., Diebold, F., and Labys, P. (2001). The distribution of realized exchange rate volatility. Journal of the American Statistical Association , 96(453):42--55

    work page 2001

  6. [6]

    Bai, Z., Chen, J., and Yao, J. (2010). On estimation of the population spectral distribution from a high-dimensional sample covariance matrix. Australian and New Zealand Journal of Statistics , 52(4):423--437

    work page 2010

  7. [7]

    and Yin, Y

    Bai, Z. and Yin, Y. (1993). Limit of the smallest eigenvalue of a large-dimensional covariance matrix. Australian and New Zealand Journal of Statistics , 21(3):1275--1294

    work page 1993

  8. [8]

    Barndorff-Nielsen, O., Hansen, P., Lunde, A., and Shephard, N. (2008). Designing realised kernels to measure ex-post variation of equity prices in the presence of noise. Econometrica , 76(6):1481--1536

    work page 2008

Show all 34 references
  1. [9]

    Barndorff-Nielsen, O., Hansen, P., Lunde, A., and Shephard, N. (2011). Multivariate realised kernels: consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading. Journal of Econometrics , 162(1):149--169

    2011

  2. [10]

    and Shephard, N

    Barndorff-Nielsen, O. and Shephard, N. (1993). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. The Annals of Probability , 64(2):253--280

    1993

  3. [11]

    Fan, J., Li, Y., and Yu, K. (2012). Vast volatility matrix estimation using high-frequency data for portfolio selection. Journal of the American Statistical Association , 107(497):412--428

    2012

  4. [12]

    Jacod, J., Li, Y., Mykland, P., Podolskij, M., and Vetter, M. (2009). Microstructure noise in the continuous case: the pre-averaging approach. Stochastic Processes and their Applications , 119(7):2249--2276

    2009

  5. [13]

    and Protter, P

    Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the euler method for stochastic differential equations. Annals of Probability , 26(1):267--307

    1998

  6. [14]

    Jing, B., Liu, Z., and Kong, X. (2017). Estimating the volatility functionals with multiple transactions. Econometric Theory , 33(1):331--365

    2017

  7. [15]

    Karoui, N. (2008). Spectrum estimation for large dimensional covariance matrices using random matrix theory. Annals of Statistics , 36(6):2757--2790

    2008

  8. [16]

    Lam, C. (2016). Nonparametric eigenvalue-regularized precision or covariance matrix estimator. The Annals of Statistics , 44(3):928--953

    2016

  9. [17]

    Lam, C., Feng, P., and Hu, C. (2017). Nonlinear shrinkage estimation of large integrated covariance matrix. Biometrika , 104(2):481--488

    2017

  10. [18]

    and Wolf, M

    Ledoit, O. and Wolf, M. (2012). Nonlinear shrinkage estimation of large-dimensional covariance matrices. The Annals of Statistics , 40(2):1024--1060

    2012

  11. [19]

    and Wolf, M

    Ledoit, O. and Wolf, M. (2017). Numerical implementation of the quest function. Computational statistics and data analysis , 115:199--223

    2017

  12. [20]

    Li, J. (2013). Robust estimation and inference for jumps in noisy high frequency data: a local-to-continuity theory for the pre-averaging method. Econometrica , 81(4):1673--1693

    2013

  13. [21]

    Liu, C., Xia, N., and Yu, J. (2016). Shrinkage estimation of covariance matrix for portfolio choice with high frequency data. arXiv:1611.0675V1

    2016

  14. [22]

    Liu, Z. (2017). Jump-robust estimation of volatility with simultaneous presence of microstructure noise and multiple observations. Finance and Stochastics , 21(1):427--469

    2017

  15. [23]

    Liu, Z., Kong, X., and Jing, B. (2018). Estimating the integrated volatility using high-frequency data with zero durations. Journal of Econometrics , 204(1):18--32

    2018

  16. [24]

    Mestre, X. (2008). Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates. IEEE Transactions on Information Theory , 54(11):5113--5129

    2008

  17. [25]

    and Zhang, L

    Mykland, P. and Zhang, L. (2009). Inference for continuous semimartingales observed at high frequency. Econometrica , 77(5):1403--1445

    2009

  18. [26]

    Silverstein, J. (1995). Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices. Journal of Multivariate Analysis , 55(2):331--339

    1995

  19. [27]

    Tao, M., Wang, Y., Yao, Q., and Zou, J. (2011). Large volatility matrix inference via combining low-frequency and high-frequency approaches. Journal of the American Statistical Association , 106(495):1025--1040

    2011

  20. [28]

    and Zou, J

    Wang, Y. and Zou, J. (2010). Vast volatility matrix estimation for high-frequency financial data. The Annals of Statistics , 38(2):943--978

    2010

  21. [29]

    and Zheng, X

    Xia, N. and Zheng, X. (2018). On the inference about the spectra of high-dimensional covariance matrix based on noisy observations-with applications to integrated covolatility matrix inference in the presence of microstructure noise. The Annals of Statistics , 46(2):500--525

    2018

  22. [30]

    Xiu, D. (2010). Quasi-maximum likelihood estimation of volatility with high frequency data. Journal of Econometrics , 159(1):235--250

    2010

  23. [31]

    Yin, Y., Bai, Z., and Krishnaiah, P. (1998). On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probability Theory and Related Fields , 78(4):509--521

    1998

  24. [32]

    Zhang, L. (2006). Efficient estimation of stochastic volatility using noisy observations: a multi-scale approach. Bernoulli , 12(6):1019--1043

    2006

  25. [33]

    Zhang, L., Mykland, P., and A\" i t-Sahalia, Y. (2005). A tale of two time scales: determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association , 100(472):1394--1411

    2005

  26. [34]

    and Li, Y

    Zheng, X. and Li, Y. (2011). On the estimation of integrated covariance matrices of high dimensional diffusion processes. Annals of Statistics , 39(6):3121--3151

    2011

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