Abstract
It is well-known from a general theorem of Hoeffding [2 ] that the limit distribution of a sequence of the so-called U-sta- tistics is normal. However, the variance of this normal distribution frequently vanishes as a consequence of improper norming. Hoeffding’s theorem is not applicable even if the statistics are elementary symmetric polynomials of independent identically distributed random variables. In fact, the limit distributions are not normal in these cases.
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References
HalÁsz, G. and Székely, G. J. (1976). On the elementary symmetric polynomials of independent random variables. Acta Math. Acad. Soi. Hungar. 28 397–400
Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 293–325
Móri, T. F. and Székely, G. J. (1979). Asymptotic behaviour of symmetric polynomial statistics. To appear in Ann. Prob.
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© 1981 Springer-Verlag New York Inc.
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Móri, T.F. (1981). On a Hoeffding-Type Problem. In: The First Pannonian Symposium on Mathematical Statistics. Lecture Notes in Statistics, vol 8. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5934-3_16
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DOI: https://doi.org/10.1007/978-1-4612-5934-3_16
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