Convergence of one-dimensional random processes
DOI:
https://doi.org/10.21754/iecos.v24i2.2005Keywords:
Wasserstein distance, Random process, Positive associateAbstract
In this paper we extensively develop some of the results obtained in reference (Cioletti et al., 2017). We use the Wasserstein distance to obtain some central limit type theorems for one-dimensional random processes having positive associated dependence.
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Ambrosio, L. (2003). Lecture Notes on Optimal Transport Problems. In: Ambrosio, L., Deckelnick, K., Dziuk, G., Mimura, M., Solonnikov, V. A., Soner, H. M., & Ambrosio, L. (Eds.), Mathematical Aspects of Evolving Interfaces (pp. 1-52). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-39189-0_1
Barlow, R. E., & Proschan, F. (1975). Statistical theory of reliability and life testing: probability models. Holt, Rinehart and Winston. https://apps.dtic.mil/sti/citations/ADA006399
Bickel, P. J., & Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. The annals of statistics, 9(6), 1196-1217. https://doi.org/10.1214/aos/1176345637
Birkel, T. (1988). Moment bounds for associated sequences. The annals of Probability, 16(3), 1184-1193. https://www.jstor.org/stable/2244116
Cioletti, L., Dorea, C. C. Y., & Vila, R. (2017). Limit Theorems in Mallows Distance for Processes with Gibssian Dependence. arXiv.
https://doi.org/10.48550/arXiv.1701.03747
Dorea, C. C., & Ferreira, D. B. (2012). Conditions for equivalence between Mallows distance and convergence to stable laws. Acta Mathematica Hungarica, 134(1-2), 1-11. https://doi.org/10.1007/s10474-011-0101-7
Esary, J. D., Proschan, F., & Walkup, D. W. (1967). Association of random variables, with applications. The Annals of Mathematical Statistics, 38(5), 1466-1474. https://doi.org/10.1214/aoms/1177698701
Gabriel, R. V. (2017). Representações gráficas para sistemas de spins com presença de campo externo: algumas relações em teoria de probabilidades [Tese para obtenção do grau de Doutor em Matemática]. Universidade de Brasília. Instituto de Ciências Exatas. Departamento de Matemática. http://icts.unb.br/jspui/handle/10482/22471
Gray, R. M. (2009). Probability, random processes, and ergodic properties. Springer. https://doi.org/10.1007/978-1-4419-1090-5
Jordan, R., Kinderlehrer, D., & Otto, F. (1998). The variational formulation of the Fokker--Planck equation. SIAM journal on mathematical analysis, 29(1), 1-17. https://doi.org/10.1137/S0036141096303359
Kantorovic ̌, L. V., Rubins ̌tei ̌n, G. S ̌. (1958). On a space of completely additive functions. Vestnik Leningrad University, 13, 52-59.
Mallows, C. L. (1972). A note on asymptotic joint normality. The Annals of Mathematical Statistics, 43(2), 508-515. https://www.jstor.org/stable/2239988
Newman, C. M. (1980). Normal fluctuations and the FKG inequalities. Communications in Mathematical Physics, 74(2), 119-128. https://doi.org/10.1007/BF01197754
Newman, C. M., & Wright, A. L. (1981). An invariance principle for certain dependent sequences. The Annals of Probability, 9(4), 671-675.
https://doi.org/10.1214/aop/1176994374
Oliveira, P. E. (2012). Asymptotics for associated random variables. Springer Science & Business Media.
Otto, F. (2001). The geometry of dissipative evolution equations: the porous medium equation. Communications in Partial Differential Equations, 26(1-2), 101-174 https://doi.org/10.1081/PDE-100002243
Pitt, L. D. (1982). Positively correlated normal variables are associated. The Annals of Probability, 10(2), 496-499. https://www.jstor.org/stable/2243445
Rachev, S. T., & Rüschendorf, L. (1998). Mass Transportation Problems: Volume I: Theory. Springer Science & Business Media.
Shorack, G. R., & Wellner, J. A. (2009). Empirical processes with applications to statistics. Society for Industrial and Applied Mathematics.
https://epubs.siam.org/doi/pdf/10.1137/1.9780898719017.bm
Sommerfeld, M., & Munk, A. (2018). Inference for empirical Wasserstein distances on finite spaces. Journal of the Royal Statistical Society Series B: Statistical Methodology, 80(1), 219-238. https://doi.org/10.1111/rssb.12236
Vaserstein, L. N. (1969). Markov processes over denumerable products of spaces, describing large systems of automata. Problemy Peredachi Informatsii, 5(3), 64-72. https://www.mathnet.ru/eng/ppi1811
Villani, C. (2003). Topics in optimal transportation. OR/MS Today, 30(3), 66-67. https://link.gale.com/apps/doc/A104669453/AONE?u=anon~6226aa1c&sid=googleScholar&xid=2585334e
Villani, C. (2009). Optimal transport: old and new. Springer. https://doi.org/10.1007/978-3-540-71050-9
Yeh, J. (2006). Real Analysis: Theory of Measure and Integration. World Scientific.
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