Summary
Suppose X 1, X 2,... are independent, identically distributed random variables, and suppose n −1/α(X1+...+Xn) converges in distribution to a symmetric stable law of index α<2. For s=1, ..., n, set
. Let Μ n be the empirical distribution of {Y ns∶ s=1, ..., n}. Then Μ n converges in distribution, but not in probability.
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Research partially supported by National Science Foundation Grant MCS 80-02535
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Freedman, D., Lane, D. The empirical distribution of the fourier coefficients of a sequence of independent, identically distributed long-tailed random variables. Z. Wahrscheinlichkeitstheorie verw Gebiete 58, 21–39 (1981). https://doi.org/10.1007/BF00536193
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DOI: https://doi.org/10.1007/BF00536193