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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Randomly sampled Riemann sums
and complete convergence
in the law of large numbers
for a case without identical distribution


Author: Alexander R. Pruss
Journal: Proc. Amer. Math. Soc. 124 (1996), 919-929
MSC (1991): Primary {60F15, 26A42; Secondary 60F10}
MathSciNet review: 1301524
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Abstract: Let the points $\{x_{nk}\}$ be independently and uniformly randomly chosen in the intervals $\left [{\frac{k-1 }{n}},{\frac{k }{n}}\right ]$, where $k=1,2,...,n$. We show that for a finite-valued measurable function $f$ on $[0,1]$, the randomly sampled Riemann sums ${\frac{1 }{n}} \sum _{k=1}^n f(x_{nk})$ converge almost surely to a finite number as $n\to \infty $ if and only if $f \in L^2[0,1]$, in which case the limit must agree with the Lebesgue integral. One direction of the proof uses Bikelis' (1966) non-uniform estimate of the rate of convergence in the central limit theorem. We also generalize the notion of sums of i.i.d. random variables, subsuming the randomly sampled Riemann sums above, and we show that a result of Hsu, Robbins and Erd\H{o}s (1947, 1949) on complete convergence in the law of large numbers continues to hold. In the Appendix, we note that a theorem due to Baum and Katz (1965) on the rate of convergence in the law of large numbers also generalizes to our case.


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Additional Information

Alexander R. Pruss
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Email: pruss@math.ubc.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03100-0
PII: S 0002-9939(96)03100-0
Keywords: Riemann sums, complete convergence, Lebesgue integral, law of large numbers, central limit theorem
Received by editor(s): May 9, 1994
Received by editor(s) in revised form: September 15, 1994
Communicated by: Rick Durrett
Article copyright: © Copyright 1996 American Mathematical Society