Summary
It is well-known that the possible limit distributions of triangular arrays (X nk )(n≧1, 1≦k≦k n )of infinitesimal random variables with independent rows are exactly the infinitely divisible distributions. In the present paper, it is shown that requiring the random variables X nk to assume only two values with prescribed probabilities p nk and 1-p nk does not narrow the class of possible limit distributions, provided that the numbers p nk satisfy 0<p nk <1 and
A similar result is obtained for limit distributions of sequences of inde-pendent random variables. As an application to probabilistic number theory, the class of possible limit distributions in the central limit theorem for additive arithmetic functions is determined.
References
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Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. Reading Mass.: Addison-Wesley 1968
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Urbanik, K.: Problem 272. Colloq. Math. 6, 336 (1958)
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Hildebrand, A. On the limit distribution of discrete random variables. Probab. Th. Rel. Fields 75, 67–76 (1987). https://doi.org/10.1007/BF00320081
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DOI: https://doi.org/10.1007/BF00320081
Keywords
- Stochastic Process
- Probability Theory
- Limit Theorem
- Number Theory
- Statistical Theory