Summary
The asymptotic behaviour of elementary symmetric polynomials S (k)n of order k, based on n independent and identically distributed random variables X 1,..., X n,is investigated for the case that both k and n get large. If \(k = o (n^{\frac{1}{2}} )\), then the distribution function of a suitably normalised S (k)n is shown to converge to a standard normal limit. The speed of this convergence to normality is of order \(\mathcal{O}(kn^{ - \frac{1}{2}} )\), provided \(k = \mathcal{O} (log^{ - 1} nlog_2^{ - 1} nn^{\frac{1}{2}} )\) and certain natural moment assumptions are imposed. This order bound is sharp, and cannot be inferred from one of the existing Berry-Esseen bounds for U-statistics. If k→∞ at the rate n 1/2 then a non-normal weak limit appears, provided the X i's are positive and S (k)n is standardised appropriately. On the other hand, if k→∞ at a rate faster than n 1/2 then it is shown that for positive X j'sthere exists no linear norming which causes S (k)n to converge weakly to a nondegenerate weak limit.
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van Es, A.J., Helmers, R. Elementary symmetric polynomials of increasing order. Probab. Th. Rel. Fields 80, 21–35 (1988). https://doi.org/10.1007/BF00348750
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DOI: https://doi.org/10.1007/BF00348750
Keywords
- Distribution Function
- Stochastic Process
- Asymptotic Behaviour
- Probability Theory
- Normal Limit