The variance of the discrepancy distribution of rounding procedures, and sums of uniform random variables

Lothar Heinrich (), Friedrich Pukelsheim () and Vitali Wachtel ()
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Lothar Heinrich: University of Augsburg
Friedrich Pukelsheim: University of Augsburg
Vitali Wachtel: University of Augsburg

Metrika: International Journal for Theoretical and Applied Statistics, 2017, vol. 80, issue 3, No 8, 363-375

Abstract: Abstract When $$\ell $$ ℓ probabilities are rounded to integer multiples of a given accuracy n, the sum of the numerators may deviate from n by a nonzero discrepancy. It is proved that, for large accuracies $$n \rightarrow \infty $$ n → ∞ , the limiting discrepancy distribution has variance $$\ell /12$$ ℓ / 12 . The relation to the uniform distribution over the interval $$[-1/2, 1/2]$$ [ - 1 / 2 , 1 / 2 ] , whose variance is 1 / 12, is explored in detail.

Keywords: Rounding residual; Euler–Maclaurin formula; Invariance principle for rounding residuals; Euler–Frobenius polynomial; Fourier-analytic approach; Primary: 62E15; 62P25; Secondary: 65B15; 60E10 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s00184-017-0609-0

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