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Journal articles on the topic 'Martingale difference sequence'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

PARCET, JAVIER, and NARCISSE RANDRIANANTOANINA. "GUNDY'S DECOMPOSITION FOR NON-COMMUTATIVE MARTINGALES AND APPLICATIONS." Proceedings of the London Mathematical Society 93, no. 1 (2006): 227–52. http://dx.doi.org/10.1017/s0024611506015863.

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We provide an analogue of Gundy's decomposition for $L_1$-bounded non-commutative martingales. An important difference from the classical case is that for any $L_1$-bounded non-commutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of non-commutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type $(1,1)$ boundedness for non-commutative martingale transforms and the non-commutative analogue of Burkholder's weak type inequality for square functions. A sequence $(x_n)_{n \ge 1}$ in a norm

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2

EL MACHKOURI, MOHAMED, and DALIBOR VOLNÝ. "ON THE CENTRAL AND LOCAL LIMIT THEOREM FOR MARTINGALE DIFFERENCE SEQUENCES." Stochastics and Dynamics 04, no. 02 (2004): 153–73. http://dx.doi.org/10.1142/s021949370400105x.

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Let [Formula: see text] be a Lebesgue space and T: Ω→Ω an ergodic measure-preserving automorphism with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on Ω with a common nondegenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A similar result is established for martingale difference sequences with densities provided the entropy is infinite. In addition, the martingale difference sequence may be chosen to be strongly mixing.

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3

Dai, Hongshuai, Tien-Chung Hu, and June-Yung Lee. "Operator Fractional Brownian Motion and Martingale Differences." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/791537.

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It is well known that martingale difference sequences are very useful in applications and theory. On the other hand, the operator fractional Brownian motion as an extension of the well-known fractional Brownian motion also plays an important role in both applications and theory. In this paper, we study the relation between them. We construct an approximation sequence of operator fractional Brownian motion based on a martingale difference sequence.

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4

Wang, Xuejun, Shuhe Hu, Wenzhi Yang, and Xinghui Wang. "Convergence Rates in the Strong Law of Large Numbers for Martingale Difference Sequences." Abstract and Applied Analysis 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/572493.

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We study the complete convergence and complete moment convergence for martingale difference sequence. Especially, we get the Baum-Katz-type Theorem and Hsu-Robbins-type Theorem for martingale difference sequence. As a result, the Marcinkiewicz-Zygmund strong law of large numbers for martingale difference sequence is obtained. Our results generalize the corresponding ones of Stoica (2007, 2011).

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5

Chen, Ying-Xia, Shui-Li Zhang, and Fu-Qiang Ma. "On the complete convergence for martingale difference sequence." Communications in Statistics - Theory and Methods 46, no. 15 (2017): 7603–11. http://dx.doi.org/10.1080/03610926.2016.1157188.

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6

Sato, Hiroshi. "Convergence of sum product of a martingale difference sequence." Hiroshima Mathematical Journal 18, no. 1 (1988): 69–72. http://dx.doi.org/10.32917/hmj/1206129861.

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7

Wang, Xue Jun, and Shu He Hu. "Complete convergence and complete moment convergence for martingale difference sequence." Acta Mathematica Sinica, English Series 30, no. 1 (2013): 119–32. http://dx.doi.org/10.1007/s10114-013-2243-8.

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8

Chen, Xia, and Hengjian Cui. "Empirical likelihood inference for partial linear models under martingale difference sequence." Statistics & Probability Letters 78, no. 17 (2008): 2895–901. http://dx.doi.org/10.1016/j.spl.2008.04.012.

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9

Rosalsky, Andrew, and Andrei I. Volodin. "On Convergence of Series of Random Elements via Maximal Moment Relations with Applications to Martingale Convergence and to Convergence of Series with p-Orthogonal Summands." gmj 8, no. 2 (2001): 377–88. http://dx.doi.org/10.1515/gmj.2001.377.

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Abstract The rate of convergence for an almost surely convergent series of Banach space valued random elements is studied in this paper. As special cases of the main result, known results are obtained for a sequence of independent random elements in a Rademacher type p Banach space, and new results are obtained for a martingale difference sequence of random elements in a martingale type p Banach space and for a p-orthogonal sequence of random elements in a Rademacher type p Banach space. The current work generalizes, simplifies, and unifies some of the recent results of Nam and Rosalsky [Teor.

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10

Chen, Yingxia. "Strong consistency of regression function estimator with martingale difference errors." Open Mathematics 19, no. 1 (2021): 1056–68. http://dx.doi.org/10.1515/math-2021-0090.

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Abstract In this paper, we consider the regression model with fixed design: Y i = g ( x i ) + ε i {Y}_{i}=g\left({x}_{i})+{\varepsilon }_{i} , 1 ≤ i ≤ n 1\le i\le n , where { x i } \left\{{x}_{i}\right\} are the nonrandom design points, and { ε i } \left\{{\varepsilon }_{i}\right\} is a sequence of martingale, and g g is an unknown function. Nonparametric estimator g n ( x ) {g}_{n}\left(x) of g ( x ) g\left(x) will be introduced and its strong convergence properties are established.

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11

Huang, Dawei, and N. M. Spencer. "On a random vibration model." Journal of Applied Probability 33, no. 4 (1996): 1141–58. http://dx.doi.org/10.2307/3214992.

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A random vibration model is investigated in this paper. The model is formulated as a cosine function with a constant frequency and a random walk phase. We show that this model is second-order stationary and can be rewritten as a vector-valued AR(1) model as well as a scalar ARMA(2, 1) model. The linear innovation sequence of the AR(1) model is shown to be a martingale difference sequence while the linear innovation sequence of the ARMA(2, 1) model is only an uncorrelated sequence. A non-linear predictor is derived from the AR(1) model while a linear predictor is derived from the ARMA(2, 1) mod

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12

Huang, Dawei, and N. M. Spencer. "On a random vibration model." Journal of Applied Probability 33, no. 04 (1996): 1141–58. http://dx.doi.org/10.1017/s0021900200100543.

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A random vibration model is investigated in this paper. The model is formulated as a cosine function with a constant frequency and a random walk phase. We show that this model is second-order stationary and can be rewritten as a vector-valued AR(1) model as well as a scalar ARMA(2, 1) model. The linear innovation sequence of the AR(1) model is shown to be a martingale difference sequence while the linear innovation sequence of the ARMA(2, 1) model is only an uncorrelated sequence. A non-linear predictor is derived from the AR(1) model while a linear predictor is derived from the ARMA(2, 1) mod

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13

Davidson, James. "The Central Limit Theorem for Globally Nonstationary Near-Epoch Dependent Functions of Mixing Processes: The Asymptotically Degenerate Case." Econometric Theory 9, no. 3 (1993): 402–12. http://dx.doi.org/10.1017/s0266466600007738.

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The central limit theorem in Davidson [2] is extended to allow cases where the variances of sequence coordinates can be tending to zero. A trade-off is demonstrated between the degree of dependence and the rate of degeneration. For the martingale difference case, it is sufficient for the sum of the variances to diverge at the rate of log n.

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14

Jena, Bidu Bhusan, and Susanta Kumar Paikray. "Statistical convergence of martingale difference sequence via deferred weighted mean and Korovkin-type theorems." Miskolc Mathematical Notes 22, no. 1 (2021): 273. http://dx.doi.org/10.18514/mmn.2021.3407.

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15

Zhao, Zhi-Wen, De-Hui Wang, and Yong Zhang. "Limit theory for random coefficient first-order autoregressive process under martingale difference error sequence." Journal of Computational and Applied Mathematics 235, no. 8 (2011): 2515–22. http://dx.doi.org/10.1016/j.cam.2010.11.004.

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16

Yang, Weiguo. "The Asymptotic Equipartition Property for a Nonhomogeneous Markov Information Source." Probability in the Engineering and Informational Sciences 12, no. 4 (1998): 509–18. http://dx.doi.org/10.1017/s0269964800005350.

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In this paper, we study the asymptotic equipartition property (AEP) for a nonhomogeneous Markov information source. We first give a limit theorem for the averages of the functions of two variables of this information source by using the convergence theorem for the martingale difference sequence. As corollaries, we get several limit theorems and a limit theorem of the relative entropy density, which hold for any nonhomogeneous Markov information source. Then, we get a class of strong laws of large numbers for nonhomogeneous Markov information sources. Finally, we prove the AEP for a class of no

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17

Ashby, Michael William, and Oliver Bruce Linton. "Do Consumption-Based Asset Pricing Models Explain the Dynamics of Stock Market Returns?" Journal of Risk and Financial Management 17, no. 2 (2024): 71. http://dx.doi.org/10.3390/jrfm17020071.

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We show that three prominent consumption-based asset pricing models—the Bansal–Yaron, Campbell–Cochrane and Cecchetti–Lam–Mark models—cannot explain the dynamic properties of stock market returns. We show this by estimating these models with GMM, deriving ex-ante expected returns from them and then testing whether the difference between realised and expected returns is a martingale difference sequence, which it is not. Mincer–Zarnowitz regressions show that the models’ out-of-sample expected returns are systematically biased. Furthermore, semi-parametric tests of whether the models’ state vari

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18

S., Elizabeth, and Nirmal Veena S. "STABILIZATION OF DISCRETE STOCHASTIC DYNAMIC SYSTEM WITH DELAY." International Journal of Current Research and Modern Education, Special Issue (August 13, 2017): 53–56. https://doi.org/10.5281/zenodo.842234.

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In this paper, we discuss the stability of stochastic type differential equations through obtaining the stability condition for the respective stochastic difference equation. The system formulation is done by considering the stochastic differential equation that describes the dynamics of single isolated neuron involving delay. Here the discretization of the stochastic differential equation is done through the Euler- Maruyama Method. And the desired stability is obtained by applying suitable assumptions and through the help of theorems. The obtained theoretical results are represented through n

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19

Li, Zhanfeng, Min Huang, Xiaohua Meng, and Xiangyu Ge. "The Limit Theorems for Function of Markov Chains in the Environment of Single Infinite Markovian Systems." Mathematical Problems in Engineering 2020 (May 5, 2020): 1–11. http://dx.doi.org/10.1155/2020/8175723.

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This paper is intended to study the limit theorem of Markov chain function in the environment of single infinite Markovian systems. Moreover, the problem of the strong law of large numbers in the infinite environment is presented by means of constructing martingale differential sequence for the measurement under some different sufficient conditions. If the sequence of even functions gnx,n≥0 satisfies different conditions when the value ranges of x are different, we have obtained SLLN for function of Markov chain in the environment of single infinite Markovian systems. In addition, the paper st

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20

Benda, Martin. "A Central Limit Theorem for Contractive Stochastic Dynamical Systems." Journal of Applied Probability 35, no. 1 (1998): 200–205. http://dx.doi.org/10.1239/jap/1032192562.

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If (Fn)n∈ℕ is a sequence of independent and identically distributed random mappings from a second countable locally compact state space 𝕏 to 𝕏 which itself is independent of the 𝕏-valued initial variable X0, the discrete-time stochastic process (Xn)n≥0, defined by the recursion equation Xn = Fn(Xn−1) for n∈ℕ, has the Markov property. Since 𝕏 is Polish in particular, a complete metric d exists. The random mappings (Fn)n∈ℕ are assumed to satisfy ℙ-a.s. Conditions on the distribution of l(Fn) are given for the existence of an invariant distribution of X0 making the process (Xn)n≥0 stationary and

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21

Chang, Bishart. "Are gold markets weak form efficient? Evidence from China, India and Russia." Sukkur IBA Journal of Management and Business 5, no. 1 (2018): 52. http://dx.doi.org/10.30537/sijmb.v5i1.189.

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The main purpose of this study is to determine the weak form efficiency of the emerging gold markets such as China, India and Russia with the special focus on testing random walks (RWS) and martingale difference sequence (MDS) hypotheses during different periods of time. This study uses biased free statistical techniques such as runs test, parametric variance ratio tests and recent modified non-parametric variance ratio tests based on ranks and signs by using daily spot gold prices from January 12, 1993 to October 28, 2016. Findings of the study suggest that Russian gold market is weak form ef

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22

Meitz, Mika, and Pentti Saikkonen. "PARAMETER ESTIMATION IN NONLINEAR AR–GARCH MODELS." Econometric Theory 27, no. 6 (2011): 1236–78. http://dx.doi.org/10.1017/s0266466611000041.

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This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a general nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first-order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. We do not require the rescaled errors to be independent, but instead only to form a stationary and ergodic martingale difference sequence. Strong consistency and asymptotic normality of the global Gaussian quasi-maximum likelihood (QML) estimator are

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23

Wang, Qiying, Yan-Xia Lin, and Chandra M. Gulati. "THE INVARIANCE PRINCIPLE FOR LINEAR PROCESSES WITH APPLICATIONS." Econometric Theory 18, no. 1 (2002): 119–39. http://dx.doi.org/10.1017/s0266466602181072.

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Let Xt be a linear process defined by Xt = [sum ]k=0∞ ψkεt−k, where {ψk, k ≥ 0} is a sequence of real numbers and {εk, k = 0,±1,±2,...} is a sequence of random variables. Two basic results, on the invariance principle of the partial sum process of the Xt converging to a standard Wiener process on [0,1], are presented in this paper. In the first result, we assume that the innovations εk are independent and identically distributed random variables but do not restrict [sum ]k=0∞ |ψk| < ∞. We note that, for the partial sum process of the Xt converging to a standard Wiener process, the condition

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24

Benda, Martin. "A Central Limit Theorem for Contractive Stochastic Dynamical Systems." Journal of Applied Probability 35, no. 01 (1998): 200–205. http://dx.doi.org/10.1017/s0021900200014789.

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If (F n ) n∈ℕ is a sequence of independent and identically distributed random mappings from a second countable locally compact state space 𝕏 to 𝕏 which itself is independent of the 𝕏-valued initial variable X 0, the discrete-time stochastic process (X n ) n≥0, defined by the recursion equation X n = F n (X n−1) for n∈ℕ, has the Markov property. Since 𝕏 is Polish in particular, a complete metric d exists. The random mappings (F n ) n∈ℕ are assumed to satisfy ℙ-a.s. Conditions on the distribution of l(F n ) are given for the existence of an invariant distribution of X 0 making the process (X n )

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25

Phillips, P. C. B. "Weak Convergence of Sample Covariance Matrices to Stochastic Integrals Via Martingale Approximations." Econometric Theory 4, no. 3 (1988): 528–33. http://dx.doi.org/10.1017/s026646660001344x.

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Under general conditions the sample covariance matrix of a vector martingale and its differences converges weakly to the matrix stochastic integral ∫01BdB′, where B is vector Brownian motion. For strictly stationary and ergodic sequences, rather than martingale differences, a similar result obtains. In this case, the limit is ∫01BdB′ + Λ and involves a constant matrix Λ of bias terms whose magnitude depends on the serial correlation properties of the sequence. This note gives a simple proof of the result using martingale approximations.

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26

Stoica, George. "Davis-type theorems for martingale difference sequences." Journal of Applied Mathematics and Stochastic Analysis 2005, no. 2 (2005): 159–65. http://dx.doi.org/10.1155/jamsa.2005.159.

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We study Davis-type theorems on the optimal rate of convergence of moderate deviation probabilities. In the case of martingale difference sequences, under the finite pth moments hypothesis (1≤p<∞), and depending on the normalization factor, our results show that Davis' theorems either hold if and only if p>2 or fail for all p≥1. This is in sharp contrast with the classical case of i.i.d. centered sequences, where both Davis' theorems hold under the finite second moment hypothesis (or less).

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27

Cullender, Stuart F., and Coenraad C. A. Labuschagne. "Unconditional martingale difference sequences in Banach spaces." Journal of Mathematical Analysis and Applications 326, no. 2 (2007): 1291–309. http://dx.doi.org/10.1016/j.jmaa.2006.03.061.

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28

Hao, Shunli. "Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences." Abstract and Applied Analysis 2013 (2013): 1–26. http://dx.doi.org/10.1155/2013/715054.

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We study the convergence rates in the law of large numbers for arrays of Banach valued martingale differences. Under a simple moment condition, we show sufficient conditions about the complete convergence for arrays of Banach valued martingale differences; we also give a criterion about the convergence for arrays of Banach valued martingale differences. In the special case where the array of Banach valued martingale differences is the sequence of independent and identically distributed real valued random variables, our result contains the theorems of Hsu-Robbins-Erdös (1947, 1949, and 1950), S

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29

McDIARMID, COLIN. "Centering Sequences with Bounded Differences." Combinatorics, Probability and Computing 6, no. 1 (1997): 79–86. http://dx.doi.org/10.1017/s0963548396002854.

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Inequalities for martingales with bounded differences have recently proved to be very useful in combinatorics and in the mathematics of operational research and computer science. We see here that these inequalities extend in a natural way to ‘centering sequences’ with bounded differences, and thus include, for example, better inequalities for sequences related to sampling without replacement.

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30

Abraham, Paul, John Alexopoulos, and S. J. Dilworth. "On the convergence in mean of martingale difference sequenceS." Quaestiones Mathematicae 23, no. 2 (2000): 193–201. http://dx.doi.org/10.2989/16073600009485968.

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31

Sun, Xichao, and Ronglong Cheng. "A Weak Convergence to Hermite Process by Martingale Differences." Advances in Mathematical Physics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/307819.

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We consider the weak convergence to general Hermite processZH,kof orderkwith indexH. By applying martingale differences we construct a sequence{ZH,kn , n=1,2,…}of multiple Wiener-Itô stochastic integrals such that it converges in distribution to the Hermite processZH,k.

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32

Giraudo, Davide. "Deviation inequalities for Banach space valued martingales differences sequences and random fields." ESAIM: Probability and Statistics 23 (2019): 922–46. http://dx.doi.org/10.1051/ps/2019016.

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We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of an orthomartingale differences random field. These inequalities can be used to give rates for linear regression and the law of large numbers.

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33

Rosalsky, Andrew, and Le Van Thanh. "Some strong laws of large numbers for blockwise martingale difference sequences in martingale type p Banach spaces." Acta Mathematica Sinica, English Series 28, no. 7 (2012): 1385–400. http://dx.doi.org/10.1007/s10114-012-0378-7.

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34

Cox, Sonja, and Mark Veraar. "Some remarks on tangent martingale difference sequences in $L^1$-spaces." Electronic Communications in Probability 12 (2007): 421–33. http://dx.doi.org/10.1214/ecp.v12-1328.

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35

Stoica, George. "Moderate deviations for bounded subsequences." Journal of Applied Mathematics and Stochastic Analysis 2006 (September 5, 2006): 1–5. http://dx.doi.org/10.1155/jamsa/2006/21782.

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We study Davis' series of moderate deviations probabilities for Lp-bounded sequences of random variables (p>2). A certain subseries therein is convergent for the same range of parameters as in the case of martingale difference or i.i.d. sequences.

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36

Ahn, Wi Chong, Bong Dae Choi, and Soo Hak Sung. "On moment conditions for supremum of normed sums of martingale differences." Bulletin of the Australian Mathematical Society 43, no. 2 (1991): 273–77. http://dx.doi.org/10.1017/s0004972700029051.

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Let {Sn, n ≥ 1} denote the partial sum of sequence (Xn) of identically distributed martingale differences. It is shown that E|X1|q (lg |X1|)r < ∞ implies E(sup((lg n)pr/q/npr/q)|Sn|p) < ∞, where 1 < p < 2, p < q, r ∈ R and lg x = max{1, log+x} For the independent identically distributed case, the converse of the above statement holds.

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37

Djellout, Hacène. "Moderate deviations for martingale differences and applications to φ -mixing sequences". Stochastics and Stochastic Reports 73, № 1-2 (2002): 37–64. http://dx.doi.org/10.1080/1045112029001/0941.

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38

Laib, Naâmane. "Exponential-type inequalities for martingale difference sequences. Application to nonparametric regression estimation." Communications in Statistics - Theory and Methods 28, no. 7 (1999): 1565–76. http://dx.doi.org/10.1080/03610929908832373.

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39

Picco, Pierre, and Maria Eulalia Vares. "A law of the iterated logarithm for geometrically weighted martingale difference sequences." Journal of Theoretical Probability 7, no. 2 (1994): 375–415. http://dx.doi.org/10.1007/bf02214275.

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40

Buldygin, V. V., and V. A. Koval. "Convergence to Zero and Boundedness of Operator-Normed Sums of Random Vectors with Application to Autoregression Processes." gmj 8, no. 2 (2001): 221–30. http://dx.doi.org/10.1515/gmj.2001.221.

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Abstract The problems of almost sure convergence to zero and almost sure boundedness of operator-normed sums of different sequences of random vectors are studied. The sequences of independent random vectors, orthogonal random vectors and the sequences of vector-valued martingale-differences are considered. General results are applied to the problem of asymptotic behaviour of multidimensional autoregression processes.

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41

Freniche, Francisco J. "Cesaro Convergence of Martingale Difference Sequences and the Banach-Saks and Szlenk Theorems." Proceedings of the American Mathematical Society 103, no. 1 (1988): 234. http://dx.doi.org/10.2307/2047557.

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42

Freniche, Francisco J. "Cesàro convergence of martingale difference sequences and the Banach-Saks and Szlenk theorems." Proceedings of the American Mathematical Society 103, no. 1 (1988): 234. http://dx.doi.org/10.1090/s0002-9939-1988-0938674-3.

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43

Shi-xin, Gan, and Qiu De-hua. "On the limiting behavior of weighted partial sums forB valued martingale difference sequences." Wuhan University Journal of Natural Sciences 7, no. 2 (2002): 133–36. http://dx.doi.org/10.1007/bf02830299.

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44

Müller, Paul F. X. "On the span of some three valued martingale difference sequences inL p (1." Israel Journal of Mathematics 60, no. 1 (1987): 39–53. http://dx.doi.org/10.1007/bf02766169.

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45

Guo, Yi, and Naiqi Liu. "Donsker-Type Theorem for Numerical Schemes of Backward Stochastic Differential Equations." Mathematics 13, no. 4 (2025): 684. https://doi.org/10.3390/math13040684.

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This article studies the theoretical properties of the numerical scheme for backward stochastic differential equations, extending the relevant results of Briand et al. with more general assumptions. To be more precise, the Brown motion will be approximated using the sum of a sequence of martingale differences or a sequence of i.i.d. Gaussian variables instead of the i.i.d. Bernoulli sequence. We cope with an adaptation problem of Yn by defining a new process Y^n; then, we can obtain the Donsker-type theorem for numerical solutions using a similar method to Briand et al.

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46

OUCHTI, L. "On the rate of convergence in the central limit theorem for martingale difference sequences." Annales de l'Institut Henri Poincare (B) Probability and Statistics 41, no. 1 (2005): 35–43. http://dx.doi.org/10.1016/j.anihpb.2004.03.003.

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47

Benaïm, Michel, and Morris W. Hirsch. "Dynamics of Morse-Smale urn processes." Ergodic Theory and Dynamical Systems 15, no. 6 (1995): 1005–30. http://dx.doi.org/10.1017/s0143385700009767.

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AbstractWe consider stochastic processes {xn}n≥0 of the formwhere F: ℝm → ℝm is C2, {λi}i≥1 is a sequence of positive numbers decreasing to 0 and {Ui}i≥1 is a sequence of uniformly bounded ℝm-valued random variables forming suitable martingale differences. We show that when the vector field F is Morse-Smale, almost surely every sample path approaches an asymptotically stable periodic orbit of the deterministic dynamical system dy/dt = F(y). In the case of certain generalized urn processes we show that for each such orbit Γ, the probability of sample paths approaching Γ is positive. This gives

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48

Cullender, Stuart F., and Coenraad C. A. Labuschagne. "Corrigendum to “Unconditional martingale difference sequences in Banach spaces” [J. Math. Anal. Appl. 326 (2007) 1291–1307]." Journal of Mathematical Analysis and Applications 338, no. 1 (2008): 751–52. http://dx.doi.org/10.1016/j.jmaa.2007.05.080.

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49

Choi, K. P., and Michael J. Klass. "Some best possible prophet inequalities for convex functions of sums of independent variates and unordered martingale difference sequences." Annals of Probability 25, no. 2 (1997): 803–11. http://dx.doi.org/10.1214/aop/1024404420.

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Lobato, I. N., John C. Nankervis, and N. E. Savin. "TESTING FOR ZERO AUTOCORRELATION IN THE PRESENCE OF STATISTICAL DEPENDENCE." Econometric Theory 18, no. 3 (2002): 730–43. http://dx.doi.org/10.1017/s0266466602183083.

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Abstract:

The problem addressed in this paper is to test the null hypothesis that a time series process is uncorrelated up to lag K in the presence of statistical dependence. We propose an extension of the Box–Pierce Q-test that is asymptotically distributed as chi-square when the null is true for a very general class of dependent processes that includes non-martingale difference sequences. The test is based on a consistent estimator of the asymptotic covariance matrix of the sample autocorrelations under the null. The finite sample performance of this extension is investigated in a Monte Carlo study.

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