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Zbl 0843.60031
Pruss, Alexander R.
Randomly sampled Riemann sums and complete convergence in the law of large numbers for a case without identical distribution. (English)
[J] Proc. Am. Math. Soc. 124, No.3, 919-929 (1996). ISSN 0002-9939; ISSN 1088-6826

Summary: Let the points $\{x_{nk}\}$ be independently and uniformly randomly chosen in the intervals $\left[ {k - 1\over n},\ {k \over n}\right]$, where $k = 1,2,\dots,n$. We show that for a finite-valued measurable function $f$ on $[0,1]$, the randomly sampled Riemann sums ${1\over n} \sum^n_{k =1} f(x_{nk})$ 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 {\it A. Bikelis'} non-uniform estimate of the rate of convergence in the central limit theorem [Litov. Mat. Sb. 6, 323-346 (1966; Zbl 0149.14002)]. 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 {\it P. L. Hsu} and {\it H. Robbins} [Proc. Nat. Acad. Sci. USA 33, 25-31 (1947; Zbl 0030.20101)] and {\it P. Erdös} [Ann. Math. Stat. 20, 286-291 (1949; Zbl 0033.29001)] on complete convergence in the law of large numbers continues to hold. In the appendix, we note that a theorem due to {\it L. E. Baum} and {\it M. Katz} [Trans. Am. Math. Soc. 120, 108-123 (1965; Zbl 0142.14802)] on the rate of convergence in the law of large numbers also generalizes to our case.

MSC 2000:: *60F15 Strong limit theorems
26A42 Ordinary integrals of functions of one real variable
60F10 Large deviations

Keywords: Riemann sums; complete convergence; Lebesgue integral; law of large numbers; central limit theorem; rate of convergence in the central limit theorem; rate of convergence in the law of large numbers

Citations: Zbl 0033.29001; Zbl 0149.14002; Zbl 0030.20101; Zbl 0142.14802

Cited in: Zbl 1147.28001

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