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), 919929
MSC (1991):
Primary {60F15, 26A42; Secondary 60F10}
MathSciNet review:
1301524
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Abstract: Let the points be independently and uniformly randomly chosen in the intervals , where . We show that for a finitevalued measurable function on , the randomly sampled Riemann sums converge almost surely to a finite number as if and only if , in which case the limit must agree with the Lebesgue integral. One direction of the proof uses Bikelis' (1966) nonuniform 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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 L. E. Baum and M. Katz, Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 120 (1965), 108123. MR 33:6679
 [2]
 A. Bikelis [= A. Bikyalis], On estimates of the remainder term in the central limit theorem, Litovski[??]i Mat. Sb. 6 (1966), 323346. (Russian) MR 35:1067
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 O. I. Klesov, Convergence of series of probabilities of large deviations of sums of independent identically distributed random variables, Ukrain. Mat. Zh. 45 (1993), 770784. (Russian) CMP 95:03
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 A. R. Pruss, Several proofs of a result concerning randomly sampled Riemann sums and complete convergence in the law of large numbers for a case without identical distribution, preprint. ftp math.ubc.ca:/pub/pruss/RandSumsBig.tex (1995).
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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/S0002993996031000
PII:
S 00029939(96)031000
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
