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A note on weak convergence


We show that in a Polish space if {P n } is a sequence of probability measures then the existence of \(\displaystyle \lim_n \int f dP_n\) for every bounded continuous function guarantees the existence of a probability P such that P n converges weakly to P.

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  • CHUNG, K.L. (1974). A course in probability theory. Academic Press, Inc. New York.

    MATH  Google Scholar 

  • DYNKIN, E.B. (1971). The initial and final behaviour of trajectories of Markov Processes. Uspehi. Math. Nauk., 26, 153–172 (English translation in Russian Math. Surveys, 26, 165–185).

    Google Scholar 

  • GNEDENKO, B.V. and KOLMOGOROV, A.N. (1968). Limit distributions for sums of independent random variables. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills., Ont.

    Google Scholar 

  • PARTHASARATHY, K.R. (1967). Probability measures on metric spaces. Academic Press, New York.

    MATH  Google Scholar 

  • RAMAKRISHNAN, S. and RAO, B.V. (1980). B-spaces are standard Borel. Ann. Probab., 80, 1191.

    MathSciNet  Article  Google Scholar 

  • VARADARAJAN, V.S. (1958a). Weak convergence of measures on separable metric spaces. Sankhyā, 20, 15–22.

    Google Scholar 

  • VARADARAJAN, V.S. (1958b). A useful convergence theorem. Sankhyā, 20, 221–222.

    MathSciNet  MATH  Google Scholar 

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Correspondence to R. V. Ramamoorthi.

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Ramamoorthi, R.V., Rao, B.V. & Sethuraman, J. A note on weak convergence. Sankhya A 74, 269–276 (2012).

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AMS (2000) subject classification

  • Primary 60F99
  • Secondary 60A10

Keywords and phrases

  • Weak convergence
  • Vitali Hahn Saks Theorem.