On local limit theorems and Blackwell's renewal theorem for independent random variables

  • Makoto Maejima


Some types of local limit theorems for independent random variables are shown and the results obtained are applied to have generalizations of Blackwell's renewal theorem.


Characteristic Function Limit Theorem Central Limit Theorem Independent Random Variable Renewal Process 


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  1. [1]
    Williamson, J. A. (1965). Some renewal theorems for non-negative random variables,Trans. Amer. Math. Soc.,114, 417–445.CrossRefMathSciNetGoogle Scholar
  2. [2]
    Maejima, M. (1972). A generalization of Blackwell's theorem for renewal processes to the case of non-identically distributed random variables,Rep. Statist. Appl. Res. JUSE,19, 1–9.MathSciNetGoogle Scholar
  3. [3]
    Cox, D. R. and Smith, W. L. (1953). A direct proof of a fundamental theorem of renewal theory,Skand. Aktuartidskr,36, 139–150.MathSciNetGoogle Scholar
  4. [4]
    Stone, C. (1965). A local limit theorem for nonlattice multi-dimensional distribution functions.Ann. Math. Statist,36, 546–551.MathSciNetGoogle Scholar
  5. [5]
    Stone, C. (1967). On local and ratio limit theorems,Proc. Fifth Berkeley Symp. Math. Statist. Prob., II, Part 2, 217–224.Google Scholar
  6. [6]
    Cramér, H. (1937).Random Variables and Probability Distributions, Cambridge University Press, Cambridge.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 1975

Authors and Affiliations

  • Makoto Maejima
    • 1
  1. 1.Keio UniversityJapan

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