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

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 Erdős (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.