International Journal of Statistical Distributions and Applications

| Peer-Reviewed |

Limit Theorems of Integrals with Respect to Vector Random Measures in Complete Paranormed Spaces

Received: 07 September 2017    Accepted: 26 September 2017    Published: 15 November 2017
Views:       Downloads:

Share This Article

Abstract

This paper studies random integral of the form, where f is a function taking value in a paranormed vector space X, and M is an independent scattered vector random measure. Random integrals of this type are a natural generalization of random series with paranormed space valued coefficients. Some limit theorems of integrals with respect to vector random measures are proved.

DOI 10.11648/j.ijsd.20170304.14
Published in International Journal of Statistical Distributions and Applications (Volume 3, Issue 4, December 2017)
Page(s) 81-86
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Paranormed Vector Space, Random Measure, Random Integral, Limit Theorem, Convergence in Probability

References
[1] S. Bochner, Stochastic Process, Annals of Mathematics, (48)1947(1), 1014-1061.
[2] F. Treves, Topological Vector Spaces, Distribution and Kernels, Academic Press, 1967.
[3] J. Rosinski, Random Integrals of Banach Space Valued Functions, Studia Mathematica, 8(1984), 15-38.
[4] P. Billingsley, Convergence of Probability Measures, John Wiley & Son, Inc. 1968.
[5] R. Zeng, the Completeness of , Journal of Mathematical Research with Applications, 15(1995)5, 40-48.
[6] L. Egghe, The Radon-Nikodym Property, densibility and Martingles in Loccally Convex Space, Pacific Journal of Mathematics, 87(1980)2, 313-322.
[7] D. H. Thang, On the Convergence of Vector Random Measures, Probability Theory and Related Fields, 88(1991), 1-16.
[8] A. N. Baushev, On the Weak Convergence of Probability Measures in Orlicz Spaces, Theory of Probability and Applications, 40(1996)3, 420–429.
[9] R. Jajte and W. D. A. Paszkiewicz, Probability and Mathematical Statistics Conditioning and Weak Convergence, Probability and Mathematical Statistics, 19(1999)2, 453-461.
[10] N. Guillotin-Plantard and V. Ladret, Limit Theorems for U-statistics Indexed by a One Dimensional Random Walk, ESAIM: Probability and Statistics, 9(2005), 98-115.
[11] H. Tsukahara, On the Convergence of Measurable Processes and Prediction Processes, Illinois Journal of Mathematics, 51(2007)4, 1231–1242.
[12] X. F. Yang, Integral Convergence Related to Weak Convergence of Measures, Applied Mathematical Sciences, 56(2011)5, 2775 – 2779.
[13] T. Grbic and S. Medic, Weak Convergence of Sequences of Distorted Probabilities, SISY 2015, Proceedings of IEEE 13th International Symposium on Intelligent System and Informatics, Sept. 2015, 307-318, Subotica, Serbia.
[14] T. E. Govindan, Weak Convergence of Probability Measures of Yosida Approximate Mild Solutions of McKean-Vlasov Type Stochastic Evolution Equations, Seventh International Conference on Dynamic Systems and Applications & Fifth International Conference on Neural, Parallel, and Scientific Computations, May 2015, Atlanta, USA.
[15] P. Puchala, Weak Convergence in L1 of the Sequences of Monotonic Functions, Journal of Applied Mathematics and Computational Mechanics 13(2014)3, 195-199.
[16] L. Meziani, A Theorem of Riesz Type with Pettis Integrals in Topological Vector Spaces, Journal of Mathematical Analysis and Applications, 340 (2008), 817–824.
[17] F. J. Pinski, G. Simpson, A. M. Stuart and H. Weber, Kullback-Leibler Ppproximation for Probability Measures on Infinite Dimensional Spaces, SIAM Journal on Mathematical Analysis, 47(2015)6, 4091-4122.
[18] W. Lohrl and T. Ripple, Boundedly Finite Measures: Separation and Convergence by an Algebra of Functions, Electronic Communication of Probability, 21(2016)60, 1–16.
[19] H. H. Wei, Weak Convergence of Probability Measures on Metric Spaces of Nonlinear Operators, Bulletin of the Institute of Mathematics Academia Sinica, 11(2016)3, 485-519.
[20] D. Dolgopyat, A Local Limit Theorem for Sums of Independent Random Vectors, Electronic Journal of Probability, 39(2017)1–15.
[21] G. Naitzat and R. J. Adler, A Central Limit Theorem for the Euler Integral of a Gaussian random field, Stochastic Processes and Their Applications, 127(2017)6, 2036-2067.
Author Information
  • School of Mathematical Sciences, Chongqing Normal University, Chongqing, China

Cite This Article
  • APA Style

    Renying Zeng. (2017). Limit Theorems of Integrals with Respect to Vector Random Measures in Complete Paranormed Spaces. International Journal of Statistical Distributions and Applications, 3(4), 81-86. https://doi.org/10.11648/j.ijsd.20170304.14

    Copy | Download

    ACS Style

    Renying Zeng. Limit Theorems of Integrals with Respect to Vector Random Measures in Complete Paranormed Spaces. Int. J. Stat. Distrib. Appl. 2017, 3(4), 81-86. doi: 10.11648/j.ijsd.20170304.14

    Copy | Download

    AMA Style

    Renying Zeng. Limit Theorems of Integrals with Respect to Vector Random Measures in Complete Paranormed Spaces. Int J Stat Distrib Appl. 2017;3(4):81-86. doi: 10.11648/j.ijsd.20170304.14

    Copy | Download

  • @article{10.11648/j.ijsd.20170304.14,
      author = {Renying Zeng},
      title = {Limit Theorems of Integrals with Respect to Vector Random Measures in Complete Paranormed Spaces},
      journal = {International Journal of Statistical Distributions and Applications},
      volume = {3},
      number = {4},
      pages = {81-86},
      doi = {10.11648/j.ijsd.20170304.14},
      url = {https://doi.org/10.11648/j.ijsd.20170304.14},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ijsd.20170304.14},
      abstract = {This paper studies random integral of the form, where f is a function taking value in a paranormed vector space X, and M is an independent scattered vector random measure. Random integrals of this type are a natural generalization of random series with paranormed space valued coefficients. Some limit theorems of integrals with respect to vector random measures are proved.},
     year = {2017}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Limit Theorems of Integrals with Respect to Vector Random Measures in Complete Paranormed Spaces
    AU  - Renying Zeng
    Y1  - 2017/11/15
    PY  - 2017
    N1  - https://doi.org/10.11648/j.ijsd.20170304.14
    DO  - 10.11648/j.ijsd.20170304.14
    T2  - International Journal of Statistical Distributions and Applications
    JF  - International Journal of Statistical Distributions and Applications
    JO  - International Journal of Statistical Distributions and Applications
    SP  - 81
    EP  - 86
    PB  - Science Publishing Group
    SN  - 2472-3509
    UR  - https://doi.org/10.11648/j.ijsd.20170304.14
    AB  - This paper studies random integral of the form, where f is a function taking value in a paranormed vector space X, and M is an independent scattered vector random measure. Random integrals of this type are a natural generalization of random series with paranormed space valued coefficients. Some limit theorems of integrals with respect to vector random measures are proved.
    VL  - 3
    IS  - 4
    ER  - 

    Copy | Download

  • Sections