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Relevant bibliographies by topics / Martingale difference sequence

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Academic literature on the topic 'Martingale difference sequence'

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Published: 3 June 2025

Last updated: 6 July 2025

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Journal articles on the topic "Martingale difference sequence"

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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 normed space $\mathrm{X}$ is called 2-co-lacunary if there exists a bounded linear map from the closed linear span of $(x_n)_{n \ge 1}$ to $l_2$ taking each $x_n$ to the $n$th vector basis of $l_2$. We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in $L_1 (\mathcal{M}, \tau)$ whose sequence of norms is bounded away from zero is 2-co-lacunary, generalizing a result of Aldous and Fremlin to non-commutative $L_1$-spaces.

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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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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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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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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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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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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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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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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. Īmovīr. ta Mat. Statist. 52: 120–131, 1995] and Rosalsky and Rosenblatt [Bull. Inst. Math. Acad. Sinica 11: 185–208, 1983, Nonlinear Anal. 30: 4237–4248, 1997].

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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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Dissertations / Theses on the topic "Martingale difference sequence"

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Paditz, Ludwig. "Beiträge zur expliziten Fehlerabschätzung im zentralen Grenzwertsatz." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-115105.

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In der Arbeit wird das asymptotische Verhalten von geeignet normierten und zentrierten Summen von Zufallsgrößen untersucht, die entweder unabhängig sind oder im Falle der Abhängigkeit als Martingaldifferenzfolge oder stark multiplikatives System auftreten. Neben der klassischen Summationstheorie werden die Limitierungsverfahren mit einer unendlichen Summationsmatrix oder einer angepaßten Folge von Gewichtsfunktionen betrachtet. Es werden die Methode der charakteristischen Funktionen und besonders die direkte Methode der konjugierten Verteilungsfunktionen weiterentwickelt, um quantitative Aussagen über gleichmäßige und ungleichmäßige Restgliedabschätzungen in zentralen Grenzwertsatz zu beweisen. Die Untersuchungen werden dabei in der Lp-Metrik, 1<br>In the work the asymptotic behavior of suitably centered and normalized sums of random variables is investigated, which are either independent or occur in the case of dependence as a sequence of martingale differences or a strongly multiplicative system. In addition to the classical theory of summation limiting processes are considered with an infinite summation matrix or an adapted sequence of weighting functions. It will be further developed the method of characteristic functions, and especially the direct method of the conjugate distribution functions to prove quantitative statements about uniform and non-uniform error estimates of the remainder term in central limit theorem. The investigations are realized in the Lp metric, 1

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Paditz, Ludwig. "Beiträge zur expliziten Fehlerabschätzung im zentralen Grenzwertsatz." Doctoral thesis, 1988. https://tud.qucosa.de/id/qucosa%3A26930.

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In der Arbeit wird das asymptotische Verhalten von geeignet normierten und zentrierten Summen von Zufallsgrößen untersucht, die entweder unabhängig sind oder im Falle der Abhängigkeit als Martingaldifferenzfolge oder stark multiplikatives System auftreten. Neben der klassischen Summationstheorie werden die Limitierungsverfahren mit einer unendlichen Summationsmatrix oder einer angepaßten Folge von Gewichtsfunktionen betrachtet. Es werden die Methode der charakteristischen Funktionen und besonders die direkte Methode der konjugierten Verteilungsfunktionen weiterentwickelt, um quantitative Aussagen über gleichmäßige und ungleichmäßige Restgliedabschätzungen in zentralen Grenzwertsatz zu beweisen. Die Untersuchungen werden dabei in der Lp-Metrik, 1<p<oo oder p=1 bzw. p=oo, durchgeführt, wobei der Fall p=oo der üblichen sup-Norm entspricht. Darüber hinaus wird im Fall unabhängiger Zufallsgrößen der lokale Grenzwertsatz für Dichten betrachtet. Mittels der elektronischen Datenverarbeitung neue numerische Resultate zu erhalten. Die Arbeit wird abgerundet durch verschiedene Hinweise auf praktische Anwendungen.<br>In the work the asymptotic behavior of suitably centered and normalized sums of random variables is investigated, which are either independent or occur in the case of dependence as a sequence of martingale differences or a strongly multiplicative system. In addition to the classical theory of summation limiting processes are considered with an infinite summation matrix or an adapted sequence of weighting functions. It will be further developed the method of characteristic functions, and especially the direct method of the conjugate distribution functions to prove quantitative statements about uniform and non-uniform error estimates of the remainder term in central limit theorem. The investigations are realized in the Lp metric, 1 <p <oo or p = 1 or p = oo, where in the case p = oo it is the usual sup-norm. In addition, in the case of independent random variables the local limit theorem for densities is considered. By means of electronic data processing new numerical results are obtained. The work is finished by various references to practical applications.

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Books on the topic "Martingale difference sequence"

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Merlevède, Florence, Magda Peligrad, and Sergey Utev. Functional Gaussian Approximation for Dependent Structures. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198826941.001.0001.

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This book has its origin in the need for developing and analyzing mathematical models for phenomena that evolve in time and influence each another, and aims at a better understanding of the structure and asymptotic behavior of stochastic processes. This monograph has double scope. First, to present tools for dealing with dependent structures directed toward obtaining normal approximations. Second, to apply the normal approximations presented in the book to various examples. The main tools consist of inequalities for dependent sequences of random variables, leading to limit theorems, including the functional central limit theorem (CLT) and functional moderate deviation principle (MDP). The results will point out large classes of dependent random variables which satisfy invariance principles, making possible the statistical study of data coming from stochastic processes both with short and long memory. Over the course of the book different types of dependence structures are considered, ranging from the traditional mixing structures to martingale-like structures and to weakly negatively dependent structures, which link the notion of mixing to the notions of association and negative dependence. Several applications have been carefully selected to exhibit the importance of the theoretical results. They include random walks in random scenery and determinantal processes. In addition, due to their importance in analyzing new data in economics, linear processes with dependent innovations will also be considered and analyzed.

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Book chapters on the topic "Martingale difference sequence"

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Zinn, Joel. "Comparison of martingale difference sequences." In Lecture Notes in Mathematics. Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/bfb0074966.

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Davis, Burgess, and Renming Song. "A Geometrical Characterization of Banach Spaces in Which Martingale Difference Sequences are Unconditional." In Selected Works of Donald L. Burkholder. Springer New York, 2011. http://dx.doi.org/10.1007/978-1-4419-7245-3_23.

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Hansen, Lars Peter, and Thomas J. Sargent. "Linear Stochastic Difference Equations." In Recursive Models of Dynamic Linear Economies. Princeton University Press, 2013. http://dx.doi.org/10.23943/princeton/9780691042770.003.0002.

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This chapter describes the vector first-order linear stochastic difference equation. It is first used to represent information flowing to economic agents, then again to represent competitive equilibria. The vector first-order linear stochastic difference equation is associated with a tidy theory of prediction and a host of procedures for econometric application. Ease of analysis has prompted the adoption of economic specifications that cause competitive equilibria to have representations as vector first-order linear stochastic difference equations. Because it expresses next period's vector of state variables as a linear function of this period's state vector and a vector of random disturbances, a vector first-order vector stochastic difference equation is recursive. Disturbances that form a “martingale difference sequence” are basic building blocks used to construct time series. Martingale difference sequences are easy to forecast, a fact that delivers convenient recursive formulas for optimal predictions of time series.

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Merlevède, Florence, Magda Peligrad, and Sergey Utev. "Moment Inequalities and Gaussian Approximation for Martingales." In Functional Gaussian Approximation for Dependent Structures. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198826941.003.0002.

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The aim of this chapter is to present useful tools for analyzing the asymptotic behavior of partial sums associated with dependent sequences, by approximating them with martingales. We start by collecting maximal and moment inequalities for martingales such as the Doob maximal inequality, the Burkholder inequality, and the Rosenthal inequality. Exponential inequalities for martingales are also provided. We then present several sufficient conditions for the central limit behavior and its functional form for triangular arrays of martingales. The last part of the chapter is devoted to the moderate deviations principle and its functional form for triangular arrays of martingale difference sequences.

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Merlevède, Florence, Magda Peligrad, and Sergey Utev. "Gaussian Approximation via Martingale Methods." In Functional Gaussian Approximation for Dependent Structures. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198826941.003.0004.

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Gordin’s seminal paper (1969) initiated a line of research in which limit theorems for stationary sequences are proved via appropriate approximations by stationary martingale difference sequences followed by an application of the corresponding limit theorem for such sequences. In this chapter, we first review different ways to get suitable martingale approximations and then derive the central limit theorem and its functional form for strictly stationary sequences under various types of projective criteria. More general normalizations than the traditional ones will be also investigated, as well as the functional moderate deviation principle. We shall also address the question of the functional form of the central limit theorem for not necessarily stationary sequences. The last part of this chapter is dedicated to the moderate deviations principle and its functional form for stationary sequences of bounded random variables satisfying projective-type conditions.

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Merlevède, Florence, Magda Peligrad, and Sergey Utev. "Moment Inequalities via Martingale Methods." In Functional Gaussian Approximation for Dependent Structures. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198826941.003.0003.

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In this chapter we establish different kinds of moment inequalities for partial sums and the maximum of partial sums of a large class of random variables, including martingale sequences, mixingales, and other dependent structures. All the bounds involve the moments of the conditional expectations of either the partial sums or the individual random variables. In most of the proofs martingale approximations are used. This method allows us to use the moment inequalities for the martingale part developed in Chapter 2. We start with a dyadic scheme useful for analysis of the variance of partial sums in the stationary setting. Then, we obtain Burkholder-type inequalities via Maxwell–Woodroofe-type characteristics and an extension of Doob’s maximal inequality for adapted sequences. A Rosenthal-type inequality for stationary sequences is also provided with bounds using conditional expectations of the partial sums. Maximal exponential inequalities are established involving either Maxwell–Woodroofe-type characteristics or the projective operators.

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Wise, Gary L., and Eric B. Hall. "Convergence in Probability Theory." In Counterexamples in Probability and Real Analysis. Oxford University PressNew York, NY, 1993. http://dx.doi.org/10.1093/oso/9780195070682.003.0010.

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Abstract In this chapter we present counterexamples related to several different concepts involving convergence in probability theory. In particular, we consider infinite products of random variables, various forms of convergence of sequences of random variables, convergence of characteristic functions, convergence of density functions and distribution functions, the central limit theorem, the laws of large numbers, and martingales.

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Lai, Tze Leung. "SOME ALMOST SURE CONVERGENCE PROPERTIES OF WEIGHTED SUMS OF MARTINGALE DIFFERENCE SEQUENCES." In Almost Everywhere Convergence II. Elsevier, 1991. http://dx.doi.org/10.1016/b978-0-12-085520-9.50020-4.

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Merlevède, Florence, Magda Peligrad, and Sergey Utev. "Linear Processes." In Functional Gaussian Approximation for Dependent Structures. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198826941.003.0012.

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Here we apply different methods to establish the Gaussian approximation to linear statistics of a stationary sequence, including stationary linear processes, near-stationary processes, and discrete Fourier transforms of a strictly stationary process. More precisely, we analyze the asymptotic behavior of the partial sums associated with a short-memory linear process and prove, in particular, that if a weak limit theorem holds for the partial sums of the innovations then a related result holds for the partial sums of the linear process itself. We then move to linear processes with long memory and obtain the CLT under various dependence structures for the innovations by analyzing the asymptotic behavior of linear statistics. We also deal with the invariance principle for causal linear processes or for linear statistics with weakly associated innovations. The last section deals with discrete Fourier transforms, proving, via martingale approximation, central limit behavior at almost all frequencies under almost no condition except a regularity assumption.

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