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}

DOI:
https://doi.org/10.1090/S0002-9939-96-03100-0

MathSciNet review:
1301524

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let the points be independently and uniformly randomly chosen in the intervals , where . We show that for a finite-valued 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) 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.

**[1]**L. E. Baum and M. Katz,*Convergence rates in the law of large numbers*, Trans. Amer. Math. Soc.**120**(1965), 108--123. 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), 323--346. (Russian) MR**35:1067****[3]**Richard Duncan and Dominik Szynal,*A note on the weak and Hsu-Robbins law of large numbers*, Bull. Polish Acad. Sci. Math.**32**(1984), 729--735. MR**86h:60036****[4]**Paul Erd\H{o}s,*On a theorem of Hsu and Robbins*, J. Math. Statist.**20**(1949), 286--291. MR**11:40f****[5]**P. L. Hsu and H. Robbins,*Complete convergence and the law of large numbers*, Proc. Nat. Acad. Sci. U.S.A.**33**(1947), 25--31. MR**8:470e****[6]**C. S. Kahane,*Evaluating Lebesgue integrals as limits of Riemann sums*, Math. Japonica**38**(1993), 1073--1076. MR**95b:28002****[7]**J. C. Kieffer and \v{C}. V. Stanojevi\'{c},*The Lebesgue integral as the almost sure limit of random Riemann sums*, Proc. Amer. Math. Soc.**85**(1982), 389--392. MR**83h:26015****[8]**O. I. Klesov,*Convergence of series of probabilities of large deviations of sums of independent identically distributed random variables*, Ukrain. Mat. Zh.**45**(1993), 770--784. (Russian) CMP**95:03****[9]**Michel Loève,*Probability theory*, Van Nostrand, New York, 1955. MR**16:598f****[10]**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:
https://doi.org/10.1090/S0002-9939-96-03100-0

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