A Note on the Convergence of Random Riemann and Riemann-Stieltjes Sums to the Integral
نویسنده
چکیده
Convergence in probability of random Riemann sums of a Lebesgue integrable function on [0, 1) to the integral has been proved. In this article we generalize the result to an abstract probability space under some natural conditions and we show L1convergence rather than convergence in probability..
منابع مشابه
Riemann-Stieltjes integrals
This short note gives an introduction to the Riemann-Stieltjes integral on R and Rn. Some natural and important applications in probability theory are discussed. The reason for discussing the Riemann-Stieltjes integral instead of the more general Lebesgue and LebesgueStieltjes integrals are that most applications in elementary probability theory are satisfactorily covered by the Riemann-Stieltj...
متن کاملRiemann-stieltjes Integral Operators between Weighted Bergman Spaces
This note completely describes the bounded or compact Riemann-Stieltjes integral operators T g acting between the weighted Bergman space pairs (A p α , A q β) in terms of particular regularities of the holomorphic symbols g on the open unit ball of C n .
متن کاملMAT125B Lecture Notes
1 Riemann integration 2 1.1 Partitions and Riemann sums . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 A criterion for integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Upper and Lower Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 The refinement of a partition . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Properties of upper an...
متن کاملOn Approximation of the Riemann–stieltjes Integral and Applications
Several inequalities of Grüss type for the Stieltjes integral with various type of integrand and integrator are introduced. Some improvements inequalities are proved. Applications to the approximation problem of the Riemann–Stieltjes integral are also pointed out.
متن کاملA new theorem on the existence of the Riemann-Stieltjes integral and an improved version of the Loéve-Young inequality
Using the notion of the truncated variation we obtain a new theorem on the existence and estimation of the Riemann-Stieltjes integral. As a special case of this theorem we obtain an improved version of the Loéve-Young inequality for the Riemann-Stieltjes integrals driven by irregular signals. Using this result we strengthen some results of Lyons on the existence of solutions of integral equatio...
متن کامل