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Journal articles on the topic 'Martingale Sequence'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Jones, Martin L. "A note on dilations and martingales." International Journal of Mathematics and Mathematical Sciences 16, no. 4 (1993): 825–27. http://dx.doi.org/10.1155/s016117129300105x.

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The purpose of this note is to investigate the effect of dilations on martingales and to give conditions under which a dilated martingale will retain the martingale property. The technique of dilating the joint distribution of a sequence of random variables has applications in optimal stopping theory and the study of prophet inequalities where martingales play a significant role.

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2

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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3

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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4

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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5

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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6

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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7

Jena, Bidu Bhusan, and Susanta Kumar Paikray. "A certain class of statistical convergence of martingale sequences and its applications to Korovkin-type approximation." Annals of the University of Craiova, Mathematics and Computer Science Series 49, no. 1 (2022): 146–57. http://dx.doi.org/10.52846/ami.v49i1.1523.

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In this paper, we investigate and study the notions of statistical product convergence and statistical product summability via deferred Cesàro and deferred Nörlund product means for martingale sequences of random variables. We then establish an inclusion theorem concerning the relation between these two beautiful and definitively useful concepts. Also, based upon our proposed ideas, we demonstrate new thoughtful approximation of Korovkin-type theorems for a martingale sequence over a Banach space. Moreover, we establish that our theorems effectively extend and improve most (if not all) of the

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8

Lynch, James D. "The Galton-Watson process revisited: some martingale relationships and applications." Journal of Applied Probability 37, no. 02 (2000): 322–28. http://dx.doi.org/10.1017/s0021900200015540.

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A martingale is used to study extinction probabilities of the Galton-Watson process using a stopping time argument. This same martingale defines a martingale function in its arguments; consequently, its derivative is also a martingale. The argumentscan be classified as regular or irregular and this classification dictates very different behavior of the Galton-Watson process. For example, it is shown that irregularity of a pointsis equivalent to the derivative martingale sequence atsbeing closable, (i.e., it has limit which, when attached to the original sequence, the martingale structure is re

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9

Yurachkivsky, Andriy. "Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale." Journal of Probability and Statistics 2011 (2011): 1–34. http://dx.doi.org/10.1155/2011/580292.

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Let for each be an -valued locally square integrable martingale w.r.t. a filtration (probability spaces may be different for different ). It is assumed that the discontinuities of are in a sense asymptotically small as and the relation holds for all , row vectors , and bounded uniformly continuous functions . Under these two principal assumptions and a number of technical ones, it is proved that the 's are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes converge in distribution to some , then a sequence () converge

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10

Lynch, James D. "The Galton-Watson process revisited: some martingale relationships and applications." Journal of Applied Probability 37, no. 2 (2000): 322–28. http://dx.doi.org/10.1239/jap/1014842539.

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A martingale is used to study extinction probabilities of the Galton-Watson process using a stopping time argument. This same martingale defines a martingale function in its argument s; consequently, its derivative is also a martingale. The argument s can be classified as regular or irregular and this classification dictates very different behavior of the Galton-Watson process. For example, it is shown that irregularity of a point s is equivalent to the derivative martingale sequence at s being closable, (i.e., it has limit which, when attached to the original sequence, the martingale structur

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11

Dai, Liang, and Mohamed-Rafik Bouguelia. "Testing Exchangeability With Martingale for Change-Point Detection." International Journal of Ambient Computing and Intelligence 12, no. 2 (2021): 1–20. http://dx.doi.org/10.4018/ijaci.2021040101.

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This work proposes a new exchangeability test for a random sequence through a martingale-based approach. Its main contributions include 1) an additive martingale which is more amenable for designing exchangeability tests by exploiting the Hoeffding-Azuma lemma and 2) different betting functions for constructing the additive martingale. By choosing the underlying probability density function of p-values as a betting function, it can be shown that, when a change-point appears, a satisfying trade-off between the smoothness and expected one-step increment of the martingale sequence can be obtained

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12

Székely, B., and T. Szabados. "Strong approximation of continuous local martingales by simple random walks." Studia Scientiarum Mathematicarum Hungarica 41, no. 1 (2004): 101–26. http://dx.doi.org/10.1556/012.2004.41.1.6.

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The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walk, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for

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13

Resnick, Sidney I., and Gennady Samorodnitsky. "Asymptotic normality of degree counts in a preferential attachment model." Advances in Applied Probability 48, A (2016): 283–99. http://dx.doi.org/10.1017/apr.2016.56.

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AbstractPreferential attachment is a widely adopted paradigm for understanding the dynamics of social networks. Formal statistical inference, for instance GLM techniques, and model-verification methods will require knowing test statistics are asymptotically normal even though node- or count-based network data are nothing like classical data from independently replicated experiments. We therefore study asymptotic normality of degree counts for a sequence of growing simple undirected preferential attachment graphs. The methods of proof rely on identifying martingales and then exploiting the mart

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14

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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15

Oulghazi, Hamid, and Fatima Ezzaki. "Strong Law of Large Numbers of Pettis-Integrable Multifunctions." Journal of Mathematics 2019 (April 15, 2019): 1–7. http://dx.doi.org/10.1155/2019/9456167.

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Using reversed martingale techniques, we prove the strong law of large numbres for independent Pettis-integrable multifunctions with convex weakly compact values in a Banach space. The Mosco convergence of reversed Pettis-integrable martingale of the form (EBnX)n≥1, where (Bn)n≥1 is a decreasing sequence of the sub σ-algebra of F is provided.

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16

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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17

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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18

Chao, Chern-Ching, Yi-Liang Chen, and Wei-Hou Cheng. "The asymptotic joint normality of the numbers of upper records, lower records and inversions in a random sequence." Journal of Applied Probability 40, no. 1 (2003): 257–63. http://dx.doi.org/10.1239/jap/1044476839.

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19

Chao, Chern-Ching, Yi-Liang Chen, and Wei-Hou Cheng. "The asymptotic joint normality of the numbers of upper records, lower records and inversions in a random sequence." Journal of Applied Probability 40, no. 01 (2003): 257–63. http://dx.doi.org/10.1017/s0021900200022385.

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20

Ciach, L. J., R. Jajte, and A. Paszkiewicz. "On the almost sure approximation of self-adjoint operators in L2 (0, 1)." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 3 (1996): 537–43. http://dx.doi.org/10.1017/s0305004100074399.

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There are several important theorems concerning the almost sure convergence of (monotone) sequences of orthogonal projections in L2-spaces. Let us mention here the martingale convergence theorems or the results on the developments of functions with respect to orthogonal systems. On the other hand every self-adjoint operator with the spectrum on the interval [0, 1] is a limit of some sequence of orthogonal projections in the weak operator topology (see [1]). This paper is devoted to a problem of approximation of a self-adjoint operator A acting in L2 (0, 1) by a sequence Pn of orthogonal projec

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21

Aihua, Fan, Wang Zhongzhi, and Ding Fangqing. "Some limit theorems of runs to the continuous‐valued sequence." Kybernetes 37, no. 9/10 (2008): 1279–86. http://dx.doi.org/10.1108/03684920810907562.

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PurposeThe purpose of this paper is to give some limit theorems on ε‐neighborhood and ε‐increasing runs of continuous‐valued dependent random sequence. In the main result no assumptions are made concerning the random variables. As corollary a result on independent case is obtained.Design/methodology/approachThe crucial part of the proof is to construct a non‐negative supper‐martingale depending on a parameter by using the moment generating function, and then applying the Doob's martingale convergence theorem.FindingsThe upper and lower bounds of the deviations from the sums of arbitrary contin

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22

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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23

Hitczenko, Paweł. "Domination inequality for martingale transforms of a Rademacher sequence." Israel Journal of Mathematics 84, no. 1-2 (1993): 161–78. http://dx.doi.org/10.1007/bf02761698.

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24

Peng, Jiangyan, and Qiying Wang. "WEAK CONVERGENCE TO STOCHASTIC INTEGRALS UNDER PRIMITIVE CONDITIONS IN NONLINEAR ECONOMETRIC MODELS." Econometric Theory 34, no. 5 (2017): 1132–57. http://dx.doi.org/10.1017/s0266466617000408.

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Limit theory with stochastic integrals plays a major role in time series econometrics. In earlier contributions on weak convergence to stochastic integrals, the literature commonly uses martingale and semi-martingale structures. Liang, Phillips, Wang, and Wang (2016) (see also Wang (2015), Chap. 4.5) currently extended weak convergence to stochastic integrals by allowing for a linear process or a α-mixing sequence in innovations. While these martingale, linear process and α-mixing structures have wide relevance, they are not sufficiently general to cover many econometric applications that have

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25

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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26

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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27

PELIGRAD, MAGDA, and COSTEL PELIGRAD. "ON THE INVARIANCE PRINCIPLE UNDER MARTINGALE APPROXIMATION." Stochastics and Dynamics 11, no. 01 (2011): 95–105. http://dx.doi.org/10.1142/s0219493711003188.

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In this paper, we establish continuous Gaussian limits for stochastic processes associated to linear combinations of partial sums. The underlying sequence of random variables is supposed to admit a martingale approximation in the square mean. The results are useful in studying averages of additive functionals of a Markov chain with normal operator.

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28

Pfeifer, Dietmar, and Ursula Heller. "A martingale characterization of mixed Poisson processes." Journal of Applied Probability 24, no. 1 (1987): 246–51. http://dx.doi.org/10.2307/3214076.

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It is shown that an elementary pure birth process is a mixed Poisson process iff the sequence of post-jump intensities forms a martingale with respect to the σ -fields generated by the jump times of the process. In this case, the post-jump intensities converge almost surely to the mixing random variable of the process.

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29

Pfeifer, Dietmar, and Ursula Heller. "A martingale characterization of mixed Poisson processes." Journal of Applied Probability 24, no. 01 (1987): 246–51. http://dx.doi.org/10.1017/s0021900200030783.

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It is shown that an elementary pure birth process is a mixed Poisson process iff the sequence of post-jump intensities forms a martingale with respect to the σ -fields generated by the jump times of the process. In this case, the post-jump intensities converge almost surely to the mixing random variable of the process.

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30

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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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

Cox, David C., and Robert P. Kertz. "Common strict character of some sharp infinite-sequence martingale inequalities." Stochastic Processes and their Applications 20, no. 1 (1985): 169–79. http://dx.doi.org/10.1016/0304-4149(85)90024-9.

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33

Astashkin, S. V. "Martingale transforms of the Rademacher sequence in rearrangement invariant spaces." St. Petersburg Mathematical Journal 27, no. 2 (2016): 191–206. http://dx.doi.org/10.1090/spmj/1383.

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34

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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35

Chauvin, B., and Rouault. "A stochastic simulation for solving scalar reaction–diffusion equations." Advances in Applied Probability 22, no. 1 (1990): 88–100. http://dx.doi.org/10.2307/1427598.

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A recent Monte Carlo method for solving one-dimensional reaction–diffusion equations is considered here as a convergence problem for a sequence of spatial branching processes with interaction. The martingale problem is studied and a limit theorem is proved by embedding spaces of measures in Sobolev spaces.

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36

Chauvin, B., and Rouault. "A stochastic simulation for solving scalar reaction–diffusion equations." Advances in Applied Probability 22, no. 01 (1990): 88–100. http://dx.doi.org/10.1017/s0001867800019340.

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A recent Monte Carlo method for solving one-dimensional reaction–diffusion equations is considered here as a convergence problem for a sequence of spatial branching processes with interaction. The martingale problem is studied and a limit theorem is proved by embedding spaces of measures in Sobolev spaces.

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37

Kolkovska, Ekaterina T. "On a stochastic Burgers equation with Dirichlet boundary conditions." International Journal of Mathematics and Mathematical Sciences 2003, no. 43 (2003): 2735–46. http://dx.doi.org/10.1155/s0161171203211121.

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We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

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38

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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39

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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40

Gouet, Raul, F. Javier López, and Gerardo Sanz. "Central limit theorems for the number of records in discrete models." Advances in Applied Probability 37, no. 3 (2005): 781–800. http://dx.doi.org/10.1239/aap/1127483747.

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Consider a sequence (Xn) of independent and identically distributed random variables taking nonnegative integer values, and call Xn a record if Xn> max{X1,…,Xn−1}. By means of martingale arguments it is shown that the counting process of records among the first n observations, suitably centered and scaled, is asymptotically normally distributed.

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41

Peng, Bo, and Zhi Hui Wu. "Pricing Option on Jump Diffusion and Stochastic Interest Rates Model." Applied Mechanics and Materials 50-51 (February 2011): 723–27. http://dx.doi.org/10.4028/www.scientific.net/amm.50-51.723.

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This paper assumed that the stock price jump process for a special kind of renewal jump process, that is incident time interval for independent and subordinate to Gamma distribution random variable sequence. We obtain the European bi-direction option pricing formulas on jump diffusion model under the stochastic interest rates by simply mathematical induce by means of martingale method.

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42

Chandra, Tapas K., and Subhashis Ghosal. "On Extensions of an Inequality of Kolmogorov." Calcutta Statistical Association Bulletin 47, no. 1-2 (1997): 1–10. http://dx.doi.org/10.1177/0008068319970101.

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The inequality of Kolmogorov (Sankhya, 1963) has been extended to a sequence of independent sub-Gaussian and other random variables. All the earlier results in the literature on this problem concerned only on the very special case of Bernoulli variables. We use martingale inequalities to establish a key result. A similar inequality is also proved for U-statistics based on exchangeable random variables.

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43

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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44

Gouet, Raul, F. Javier López, and Gerardo Sanz. "Central limit theorems for the number of records in discrete models." Advances in Applied Probability 37, no. 03 (2005): 781–800. http://dx.doi.org/10.1017/s0001867800000471.

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Consider a sequence (X n ) of independent and identically distributed random variables taking nonnegative integer values, and call X n a record if X n &gt; max{X 1,…,X n−1}. By means of martingale arguments it is shown that the counting process of records among the first n observations, suitably centered and scaled, is asymptotically normally distributed.

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45

Yakubovich, Yuri, Oleg Rusakov, and Alexander Gushchin. "Functional Limit Theorem for the Sums of PSI-Processes with Random Intensities." Mathematics 10, no. 21 (2022): 3955. http://dx.doi.org/10.3390/math10213955.

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We consider a sequence of i.i.d. random variables, (ξ)=(ξi)i=0,1,2,⋯, Eξ0=0, Eξ02=1, and subordinate it by a doubly stochastic Poisson process Π(λt), where λ≥0 is a random variable and Π is a standard Poisson process. The subordinated continuous time process ψ(t)=ξΠ(λt) is known as the PSI-process. Elements of the triplet (Π,λ,(ξ)) are supposed to be independent. For sums of n, independent copies of such processes, normalized by n, we establish a functional limit theorem in the Skorokhod space D[0,T], for any T>0, under the assumption E|ξ0|2h<∞ for some h>1/γ2. Here, γ∈(0,1] reflects

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46

Gouet, Raúl, F. Javier López, and Miguel San Miguel. "A martingale approach to strong convergence of the number of records." Advances in Applied Probability 33, no. 4 (2001): 864–73. http://dx.doi.org/10.1239/aap/1011994033.

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Let (Xn) be a sequence of independent, identically distributed random variables, with common distribution function F, possibly discontinuous. We use martingale arguments to connect the number of upper records from (Xn) with sums of minima of related random variables. From this relationship we derive a general strong law for the number of records for a wide class of distributions F, including geometric and Poisson.

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47

Gouet, Raúl, F. Javier López, and Miguel San Miguel. "A martingale approach to strong convergence of the number of records." Advances in Applied Probability 33, no. 04 (2001): 864–73. http://dx.doi.org/10.1017/s000186780001123x.

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Abstract:

Let (X n ) be a sequence of independent, identically distributed random variables, with common distribution function F, possibly discontinuous. We use martingale arguments to connect the number of upper records from (X n ) with sums of minima of related random variables. From this relationship we derive a general strong law for the number of records for a wide class of distributions F, including geometric and Poisson.

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48

JONES, ROGER L., ROBERT KAUFMAN, JOSEPH M. ROSENBLATT, and MÁTÉ WIERDL. "Oscillation in ergodic theory." Ergodic Theory and Dynamical Systems 18, no. 4 (1998): 889–935. http://dx.doi.org/10.1017/s0143385798108349.

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Abstract:

In this paper we establish a variety of square function inequalities and study other operators which measure the oscillation of a sequence of ergodic averages. These results imply the pointwise ergodic theorem and give additional information such as control of the number of upcrossings of the ergodic averages. Related results for differentiation and for the connection between differentiation operators and the dyadic martingale are also established.

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49

Srivastava, H. M., Bidu Bhusan Jena, and Susanta Kumar Paikray. "Deferred Cesaro statistical convergence of martingale sequence and Korovkin-type approximation theorems." Miskolc Mathematical Notes 23, no. 1 (2022): 443. http://dx.doi.org/10.18514/mmn.2022.3624.

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50

Melamed, Benjamin, and Ward Whitt. "On arrivals that see time averages: a martingale approach." Journal of Applied Probability 27, no. 2 (1990): 376–84. http://dx.doi.org/10.2307/3214656.

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This paper is a sequel to our previous paper investigating when arrivals see time averages (ASTA) in a stochastic model; i.e., when the steady-state distribution of an embedded sequence, obtained by observing a continuous-time stochastic process just prior to the points (arrivals) of an associated point process, coincides with the steady-state distribution of the observed process. The relation between the two distributions was also characterized when ASTA does not hold. These results were obtained using the conditional intensity of the point process given the present state of the observed proc

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